What we perceive as "flavor" often comes from a lot of aromatic and volatile components that we smell. We smell them because they are volatile, which means that they tend to evaporate off food (if they are small molecules) or tend to be carried off of food (for larger molecules). Aside from the basic sweet, sour, bitter, salt, and umami notes, the rest of the complexity of flavor comes from smell and volatile flavor molecules.
If the only thing that evaporated when reducing a sauce was "pure water" in the form of steam, you wouldn't be able to smell it. Yet, when you cook a stock or sauce for a long time, the whole kitchen and house often smells with great aromas.
More aromas in your house means more flavor lost from the sauce. So, if you do add water back into a sauce, your sauce will often taste somewhat more diluted. Or, more likely, it will taste "unbalanced," since the smaller flavor molecules tend to escape faster and easier in evaporation. When we cook down a sauce and concentrate it, we're used to the way the flavors change. But when you dilute it again, the flavors may not have the same proportion as the original.
All of that said, the magnitude of this effect varies a lot depending on the sauce and types of flavors in it. If you over-reduce slightly and just add a bit of water back in, it's not likely to make a huge difference. But if you took a giant pot of flavorful meat stock (which tastes fine as-is), then reduced it down to a demiglace, and then added water back in to try to re-create the original stock, it definitely would taste very different and many flavor notes would be significantly weaker or lost altogether.
As Harold McGee notes in On Food and Cooking (pp. 600-601):
Slowly simmered until it's reduced to a tenth its original volume,
stock becomes glace de viande, literally "meat ice" or "meat glass,"
which cools to a stiff, clear jelly. Glace has a thick, syrupy,
sticky consistency thanks to its high gelatin content, about 25%, an
intensely savory taste thanks to the concentrated amino acids [large,
non-volatile molecules], and a rounded, mellow, but somewhat flat
aroma thanks to the long hours during which volatile molecules have
been boiled off or reacted with each other. ... Intermediate between
stock and glace is demi-glace or "half-glace" which is stock
simmered down to 25-40% of its original volume. ... [I]t also has
the advantage of sparing some of the stock's flavor from being boiled
off. ... Demi-glace is the base for many classic French brown
sauces.
Or, listen to James Peterson, author of Sauces: Classical and Contemporary Sauce Making (3rd ed., p. 91):
Do not overreduce. Stocks are often reduced to concentrate their
flavor and to give them an appetizing, light, syrupy texture.
Although reduction is an almost essential technique for converting
stocks into sauces, much of the delicacy and flavor of meats is lost
if reduced too long. Many of the flavors contained in stock are
aromatic and evaporate when simmered over a prolonged period, leaving
a flat taste.
In a later chapter (p. 112), he adds: "Stocks and sauces that have been overly reduced often have a flat, cooked taste that must be offset with more assertive flavors." As Escoce's answer points out, when you reduce a sauce significantly, it may "overcook" the flavors. Some of that may be due to browning (Maillard) reactions, but prolonged cooking will also break down some large flavor molecules, again lessening certain flavor notes and altering the balance. In tomato sauce in particular, you may also break down thickening components (specifically pectin) by overcooking, which will make your sauce more watery. For vegetable-based sauces, this can often be prevented by boiling the sauce briefly to deactivate enzymes before simmering to reduce.
Because of the flavor changes and loss during reduction, many cuisines around the world have a tradition of adding some "fresh" ingredients near the end of cooking and/or when serving, such as fresh herbs, a spritz of fresh juice, or some chopped fresher vegetables. The flavors we associate with "fresh" are often the most volatile ones, given off early when a food is first cut or cooked. Adding them at the end can give a depth of flavor that wouldn't be preserved in a long-cooked reduced sauce.
If you do end up with an overreduced sauce, the flavor is often best salvaged by adding some flavorful new ingredients (as in the Peterson quotation above) or at least adding some more of the "base" back in, such as more stock in a reduced stock or more tomatoes/tomato paste in a tomato sauce rather than plain water.
So, how do you minimize flavor loss when reducing? It's tough, but it depends on whether more problems are created in flavor loss through evaporation or through chemical changes in the remaining liquid (like flavor molecules breaking down). In most cases, the evaporation is actually a greater concern when making a sauce, so simmering slowly in a tall pot can help. The more violent the surface of the liquid is, the more the larger flavor molecules are thrown off and can escape: thus cooking down a sauce at a full boil is usually never good for flavor. (See above link for more details.) A taller pot will allow more condensation to occur before the steam leaves the pot, making it more likely that bigger flavor molecules will not have sufficient kinetic energy to escape the pot and/or will be caught in the sort of "rain shower" of condensation and be returned to the liquid. (This is the same process used in fractional distillation.) Of course, these also slow down the reduction process, so it's a trade-off in flavor vs. time.
