Summary: All materials are different, and copper is no exception. It has some unique thermal properties that may be desirable for some applications. But other combinations of materials (particularly aluminum) within a modern stainless pan can have other unique advantages that come close to -- and in some ways exceed -- copper's properties.
(For a detailed comparison of the thermal properties, see points (4), (5), and (6) below.)
First, a clarification: Despite the mention of "pure stainless steel" in the first sentence of the question, the question is apparently NOT about comparing "pure" stainless pans (based on subsequent comments), but rather whatever modern pans are basically stainless on the cooking surface and possibly the stove surface as well. (Pure stainless is a terrible conductor of heat and is rarely used these days on its own in cookware.)
This makes a significant difference because design factors of individual pans play a much greater role in comparisons than the outer surface of the pans alone. Do the stainless pans contain an aluminum disk or "core" (or one with copper or silver or something else)? What is the thickness of each of these materials in the pan, and how are they deployed? Shape, size, and other factors will play smaller roles.
All of these differences in design make it quite difficult to evaluate "practical" evidence, since it's hard to define what "pure stainless steel" pan would be "equivalent" to a particular copper pan. One of exactly the same dimensions? One of the same diameter and design but with a different thickness to make it the same weight? One of the same diameter but with a thickness required to attain a similar heat conductivity profile on the cooking surface? For this reason, I would say even my own measurements that were quoted in another response should be treated as "anecdotal" evidence. They only prove that differences existed on my particular pans. I mainly did them to prove that cast iron wasn't as "even" a cooking surface as is commonly asserted. I don't claim that they should be taken as absolute evidence for whether any copper pan is "better" in heat conduction than some other material.
All the other "experiments" that are quoted in other responses (such as McGee's tests already mentioned or these, for another example) can be criticized similarly. It's not enough to say "the pans are roughly equal in size and shape." Details in thickness and design can make a huge difference. Unless we establish clear criteria for what constitutes "equivalent" pans for the comparison, the measurements will really only just compare performance from one pan to another, not from copper pans (or other materials) in general to other things.
What I do take my measurements to be evidence for is that -- at least in common scenarios -- the conductivity numbers which are quoted for various materials do seem to roughly line up with practical evidence. Cast iron is not, for example, magically "very even" despite its terrible conductivity numbers. Copper and aluminum, on the other hand, appear to be more even. And copper appears to have a slight thermal advantage in my anecdotal test, which again accords with theoretical predictions.
With the knowledge that those conductivity numbers do seem to mean something, the question then becomes: Can we design a pan involving a clad stainless steel design (perhaps with no copper?) that has similar characteristics to a traditional copper pan? (And are such pans available?)
And the answer is: sort of.
Part of the problem is determining what "better" means in this question. I'll consider a number of desirable factors in turn: (1) durability, (2) design, (3) maintenance and cleaning, (4) evenness in heating, (5) responsiveness, (6) ability to absorb and radiate heat, and (7) cost. While the question focuses on thermal abilities, the others are explicitly or implicitly invoked in by the question.
In the following discussion, I will assume the most common type of modern stainless steel pans enhanced by an aluminum disk or "core." A few high-end manufacturers have produced lines with thick layers of copper (and sometimes silver, an even better conductor) in the interior, but these effectively behave like copper pans, because they are actually mostly made out of copper. (Note that this "Copper Core" line is often cited as an example, but the manufacturer has not released details about the thickness of the copper in this pan, so it may in fact be a mostly aluminum core.)
(1) Durability -- Modern stainless steel pans, even relatively cheap ones, are generally durable. While it is possible for stainless pans with layers of different materials (e.g., aluminum) to warp or separate, it's pretty rare except in really cheap pans. Copper pans often have a reputation for being more durable, but that's probably because the few manufacturers left tend to be high-end quality producers. The most common failure point in pans is the riveting or welding between handle and pot, but this is a design and manufacture issue, not one depending on pan material.
Thin copper does have anecdotal evidence of warping on occasion, just as thin pure aluminum pans do. In most cases, minor warping can be corrected in copper with simple tools. Stainless pans warp very rarely. But when they do, they can be nearly impossible to fix, particularly if the warping has also led to separation in the layers of aluminum or other materials. High-quality thick pans of either sort should be very durable, however.
(2) Design -- Obviously both types of pans are available in a variety of designs, but the materials place certain constraints. The main one is the density of copper, which is roughly 3.3 times that of aluminum. Thus, copper pans of equivalent thickness to aluminum or stainless/aluminum pans will generally be much heavier. Professional line copper pans (usually 2.5 to 3mm thick) have roughly similar weight to cast iron pans of similar dimensions. For some people, the inconvenience of handling heavy pans may not be worth the added thermal properties.
The weight of copper also means that traditional copper pans favor designs with cast iron handles and heavy rivets which are able to take the weight. Lighter stainless/aluminum hybrids can use other handles designed so they won't heat up excessively, while the cast iron handles on copper will get very hot during long cooking (requiring a pot holder, towel, or perhaps a silicone pot handle cover). Some thinner copper lines (1-2mm thick) have other materials for handles, but almost all will heat up significantly during cooking.
In general, copper pan designs tend to follow very traditional models, so people who want more variety in design or ergonomics may have to look to stainless pans.
(3) Maintenance and Cleaning -- Copper has a reputation for being high-maintenance. It is true that if you want to keep you pans gleaming bright with a mirror finish, you'll have to polish them on a regular basis (probably at least every few times you use the pans). However, the dark "patina" that gradually oxidizes the copper exterior is not harmful to the pan's performance (and in fact will help it, sometimes significantly, see (6) below).
If you think you need bright and shiny pans to hang on your wall and show off to visitors, either don't actually cook in your pans or be prepared to polish them all the time. Many serious cooks come to realize that the darkened exteriors are just normal; others polish only a couple times per year or only when a significant undesirable stain occurs.
Traditional copper pans were generally lined with tin, which will gradually wear down over time and require retinning. Also, tin melts at low temperatures, making the pans unusable for high-temperature searing. Heating a dry pan could even result in bubbling or melting of the tin coating, though this usually requires particular neglect. Eventually, the tin will need to be redone, and there are only a limited number of shops capable of doing this today, so it may require shipping the pans away for weeks or months. (It can also be expensive.) For most home cooks, this probably would only need to be done once per decade or so with high-use pans, but it is a significant maintenance issue to consider.
For the past few decades, however, the more common choice for home cooks has been to line copper with a very thin layer of stainless steel (usually only 0.2mm thick or so). This doesn't really change the performance of the pan, and the steel lining is permanent and can be heated to higher temperatures. (Some high-end copper pans also are available with silver linings. These are obviously generally quite pricey, but silver has both a higher conductivity than copper and a much higher melting point than tin, making the pans marginally better than stainless-lined copper.)
Stainless steel pans, on the other hand, are generally low maintenance and can often be put in the dishwasher, though high-end pans with many layers of various materials in the base often suggest handwashing when feasible. Food might have a greater tendency to stick in pans with worse thermal properties, making cheaper stainless pans potentially a little harder to clean after cooking. But neither good stainless/aluminum nor good copper should have this problem.
(4) Evenness in Heating -- Here is where we get to the most commonly cited advantages of copper. It is true that copper's conductivity is roughly 70% higher than aluminum, which most people take as evidence that copper has a significant culinary advantage.
But conductivity is not always the most helpful measure, since it is an abstract measurement of heat conduction in one dimension. That makes it somewhat useful as an estimate for whether a material will carry any significant heat radially outward beyond the burner (e.g., when a large pan is on a small burner, or up the sides of a pan), but it does not take into account how much heat the pan can hold onto in a particular volume of the pan. The latter (volumetric heat capacity) is also important to consider if you want a really even pan without hot spots. Copper still has an advantage over aluminum by volume (about 40% more heat capacity), but aluminum has an advantage by weight (since aluminum is much less dense, it has about 2.3 times the heat capacity by weight). This last fact means that is possible to get a lighter aluminum pan that will hold heat better compared to copper, even if it is somewhat thicker.
But don't take my word for it. The authors of the recent book Modernist Cuisine designed a detailed model and performed experiments to test evenness of different materials. They found that you can achieve the same evenness of a pro line of 2.5mm copper pans by using a 7mm aluminum pan. In fact, since evenness really is only dependent on thickness, you could make a really slow material be just as even as long as it's thick enough: a 2.75-inch thick slab of stainless steel would also be as "even" as 2.5mm of copper. Unfortunately, a pan made of steel that thick (22 times the thickness of copper pans) would likely weigh hundreds of pounds, and it would have other issues (see next item). But it would be just as "even" as 1/8"-thick copper.
If your only goal is evenness and no hot spots, you just need to make you pan thicker, whatever material it is made out of. In fact, you could achieve the same goal by putting a metal slab (or a heat diffuser disk) made of copper or aluminum on your burner, and putting your pan on top of that. Whatever material you are using, the heat will be more "even" overall, and this may be a useful trick when you need to put a very wide pan on a very small burner. The Modernist Cuisine folks actually concluded that your heat source was much more important than your pan design in terms of getting evenness for this reason.
(5) Responsiveness -- In a slow-cooking stock pot or a large soup pot filled with chili, evenness is probably top priority. (Hence, to answer the specific question about a stock pot, I'd recommend buying a pot with a thick bottom and perhaps getting a heat diffuser, if evenness is the main concern.)
But evenness is not everything. Some have misinterpreted the data from the Modernist Cuisine findings cited above to conclude that there's no reason to spend extra money for copper or other expensive pans (see here, for example). Their solution is just to buy a "thick pan" no matter what the material, and if you pans are still bad, put a thick aluminum plate over your burner.
But this is a flawed conclusion, since a thick aluminum plate on your stove effectively turns a high-end performance gas stove into a cheap electric in terms of how fast it can change heat, for example.
People don't pay for expensive cookware only because it heats evenly. You could do that with any material. The problem is balancing evenness with responsiveness. If you turn up the heat on a cast iron pan too much, for example, it will continue to heat your food long after you turn off the burner. If you're not careful, you can easily burn food this way. For even more sensitive dishes (those involving eggs, or milk, or thick sauces that could stick and burn, or when cooking sugar or chocolate, etc.), you want the pan to stop heating immediately when you hit a narrow temperature range. This is very difficult to calibrate with a pan with a high heat capacity, like an excessively thick cheap pan. A thick metal plate on your stovetop will do the same.
For responsiveness, it's not just about conductivity or heat capacity. It's the relationship between these two concepts -- heat conduction and heat retention -- which will determine whether those hot spots will ever even out in a given pan. For that, diffusivity is the most appropriate metric, since it combines both of these: it effectively measures the rate at which temperature evens out throughout the pan, whether hotspots or coldspots or a "new wave" of heat when the burner it turned up. Here, copper has only a 20-25% lead over aluminum.
Because copper's diffusivity isn't that much higher, we can design a pan that will shift temperature almost as fast with aluminum. Actually, we can easily design a pan that will change temperature very quickly: just make it ridiculously thin. To take this to an extreme, imagine cooking on a piece of aluminum foil. The foil will adjust temperature almost immediately, but it has no heat capacity, making it difficult to convey an even heat. Hot spots will be anywhere the heat source is uneven.
On the other hand, if we make the 7mm thick aluminum pan above (presumably covered in stainless), which could have the evenness of 2.5mm copper, the 7mm aluminum/stainless pan would have almost double the heat capacity of the 2.5 copper pan.
This translates into significantly less responsiveness, because when you turn the burner off underneath the 7mm pan, it has twice as much heat to dissipate from the pan. If your egg-based sauce is already starting to set too quickly, you might have a problem with the aluminum.
This is why most professional copper pans have a maximum thickness of 2.5 to 3mm. Unless you're making a stockpot or something else where you actually don't want the pan to change temperature quickly, a thicker copper would be counterproductive. It would make the cookware not only heavier, but less responsive. Through trial and error over the centuries, copper manufacturers seem to have discovered the "sweet spot" for copper thickness is about 2mm to 3mm for most cooking. Thinner, and the pan doesn't have enough to even out the hot spots; thicker, and it doesn't respond. (The 1.5mm lines you often see in high-end cooking shops are too thin: top restaurants use them to serve food in for their appearance, not for serious cooking. If you are willing to go with the lower heat capacity and evenness of 1.5mm copper, you might as well go with a thicker, cheaper, and lighter aluminum pan, which might be as even, and almost as responsive.)
Similarly, manufacturers of pans made of aluminum clad in stainless have realized that 7mm is just too thick for all-purpose cookware, so you rarely see aluminum that thick except in large pots (usually pure aluminum commercial heavy pots, not necessarily with stainless).
Instead, most quality aluminum/stainless lines use a disk or core that is 3 to 5mm thick, which makes the pans closer to the responsiveness of thick copper, but not quite as even.
So, you can't actually get an exact match to the thermal properties of copper with an aluminum/stainless pan. But you can get close. There's nothing magical about the "copper sweet spot" that says it is always the best. If you want a pan with a slightly faster response than copper with better retention (and much lighter in weight), but not quite as even, you can get that in a good aluminum/stainless pan.
(6) Ability to Absorb and Radiate Heat -- This one is often neglected in comparisons, but it actually can have significant effects. All of the properties discussed so far have to do with how fast heat moves inside the pan. But cooking also requires heat to be transferred into the pan, and then out from the pan into the food. In traditional gas cooking, most of this heat is transferred via conduction (materials in direct contact) and convection (air currents around the pan). However, there is a third method of heat propagation through radiation, which is particularly relevant to electric stoves (and especially many of the glass-top ones).
Darker pans absorb heat better and give it off faster. It's the reason why dark baking pans brown cakes faster, and why cast iron browns food faster than some other materials. The property that measures this is known as emissivity.
This is why you actually are harming your copper pans by shining them too much. A shiny pan reflects heat just as it reflects visible light. Stainless steel pans have a natural shiny finish that won't really go away (though it may get dull over time). If your heating elements use a lot of radiant heat, stainless pans will always reflect a lot of that heat and not work as efficiently.
Copper, on the other hand, will gradually darken in color with use, thereby increasing the rate at which the pan absorbs heat (as well as how fast it dissipates it when removed from heat). Again, it's mostly relevant to cooking situations that involve a lot of radiant heat, but the effects can be noticeable and significant.
(Want "practical" evidence again? See this thread where a guy acquired a brand-new copper pan with a mirror finish and found that he couldn't even boil water quickly on his glass-top electric stove. He thought it was defective. After a week of use, the copper dulled and darkened, and it actually responded as it should.)
(7) Cost -- This is the last -- and most obvious -- factor. Copper costs a lot. Is it worth it? Only if you want the particular characteristics mentioned above: a specific combination of evenness and fast responsiveness. This is mostly useful for some particularly sensitive dishes that require a pan that is both even and can "stop on a dime" in heating when necessary. Even many of those things can be prepared in a double-boiler or something instead, as long as they don't require heating above boiling.
If you want mainly evenness for some applications (like a stockpot?), just go with a thicker pan. If you want more responsiveness, go with a thinner pan. (Years ago, some cast iron was significantly thinner than today, because it was meant for all-purpose cooking. Even cast iron can seem relatively responsive if it's thin enough -- but it will have major hotspots.)
Copper arguably has a superior balance, but in many cases, an aluminum/stainless combo will do just fine. In fact, as I pointed out, in some cases it might be better if you want somewhat higher responsiveness in a lighter pan. (This is why I don't recommend a 1.5mm line of copper unless you're just buying it for the looks: it's not as durable, and it can basically be imitated in performance with a cheap aluminum pan. If you want to buy copper, the average balance for most pans is around 2.5mm.)
One final factor to consider in cost is efficiency. A pan that can conduct heat faster will convey more heat through it rather than reflecting the heat around it. (And, as I noted above, the dark finish of well-worn copper is even better at this.) With most cookware, a lot of the heat from the stove is reflected around the pan and warms the kitchen, rather than the food. Copper will transfer more energy directly to the food. Over time, this can save a small amount of money in energy savings. Compared to a well-designed aluminum pot, perhaps not a lot, but compared to less efficient cookware, it may save you a few dollars per month in energy if you cook regularly. Over a few decades, a few dollars per month can accumulate to hundreds or thousands of dollars.
Is this an argument to buy copper? Not necessarily. But it makes it seem more affordable. In fact, it's really an argument for any more efficient and durable cookware, even it is costs a little more. Similarly, a better aluminum/stainless pan might cost more, but if it has a better balance of aluminum thickness, is more durable (so it can last many years), and is reasonably energy efficient, it might also pay for itself over time in energy savings.