An old question, but I have a few things to add and clarify.
The question is a bit like preparing to "upgrade" to a new car, and asking people: "Should I buy the Mustang or the giant Ford pickup truck?" There's really not a good answer to that question, since the items in question are at different ends of the car spectrum and they have vastly different strengths and weaknesses. The short version of my answer is thus: "Just buy the right tool for the job."
I've already basically compared cast iron and copper in response to another question (see here), and I've documented my own detailed comparisons of cast iron and copper heat conduction performance elsewhere. For myself, I'd consider copper much more versatile as "all-purpose" cookware. And if you don't mind the multicolored "patina" of well-used copper and get stainless linings (as opposed to more traditional tin that will wear over time), the maintenance is less than cast iron.
Contrary to information in some of the other answers, I've rarely heard of thick copper warping or bending significantly, and even if it does, it can be generally beaten back into shape. Cast iron, on the other hand, is basically impossible to reshape if it warps, and I have had that happen myself; I don't think it's common, though.
Related to that issue, I would say that copper is only worthwhile if you're really going for the "good stuff," i.e., professional grade copper that is at least 2.5 to 3mm thick (which doesn't generally warp without significant abuse). The thinner copper pots, which are often marketed as cheaper alternatives at fancy cooking stores, are fine, but generally not worth the extra expense. Top restaurants use those thin copper pots to serve food in, not generally for serious cooking. If you are tempted by that 1.5mm copper (sometimes even thinner), you're mostly paying a premium for the looks. You could get basically the same performance out of a heavy-bottomed thick aluminum pan (or a stainless steel pan with a thick aluminum layer), usually for a lot cheaper. Aluminum has a lower conductivity than copper, but similar diffusivity, which means if you add extra thickness, it can perform about as well as equivalent (thinner) copper. Also, most pans with a copper clad base or a copper "layer" are marketing ploys -- the copper layer is often too thin to make any significant difference. Always ask about these, and if the copper layer isn't at least 2mm thick or so, it's probably not worth the premium price.
So now we get to the real issue everyone argues about: price. Copper seems like a luxury, and admittedly, the performance gains over modern aluminum (often coated with stainless) aren't huge. My one final comment on that issue is: don't discount potential energy savings. Copper's superior conductivity means that it will absorb heat coming from your burner more effectively, so you generally cook over lower heat to get the same results. This also means that there is less wasted heat released into your home. (This is particularly noticeable in the summer when your stove "fights" your air conditioning.)
There are reports of people who have measured this effect by switching to copper cookware, where cooking is their only use of gas in their home, and they've noticed significant decreases in gas use. I think for most people the difference is pretty small, since most people don't spend a lot of money just to power their stoves. But, based on my own usage and few experiments (like difference in times to boil water or do other heating tasks with the same heat and different pans), I'd put my energy savings on the order of at least $5/month or so. That doesn't sound like a lot, but that's $600/decade. Professional copper cookware is built to last, and people pass it down through generations. Given rising energy costs all the time, it's possible you could make an entire set of professional copper cookware pay for itself in a few decades of use, just in energy savings alone. At a minimum, you will save something in energy costs because copper is simply more efficient -- which means copper pans aren't really as expensive in the long run, taking all factors into consideration.
Of course, there are other ways to save energy and potentially get quicker response, like buying an induction stove (which, by the way, won't work with copper, but will with cast iron). In that case, you're probably likely to pay a premium of several hundred to a couple thousand dollars for the relatively new technology, and ranges generally will fail in 10 years or so (15 if you're lucky). So it probably ends up being a similar cost to buying a set of copper pans.
Overall, though, I think the best advice is to buy a sample and try. Buy one copper pan/pot, and use it for as many tasks as you can think of. (It's pricey, but you really can't evaluate how it works without trying it. I cook quite a bit, and I was somewhat shocked the first time I used a real pro-grade copper pot.) Also, buy a pan of whatever kind of cast iron, and do similar things. Figure out what you like and what works best for you, and then buy more of whatever it is for the cooking you do.