While we'll all say it comes down to personal preference, there are a couple of things that aging certainly does to tobacco. The first thing to note is that many plants including tobacco contain tannins. It's the chemical that makes your lips pucker when you eat cranberries or an unripened raspberry. They're these long chains of molecules that exist throughout most plants. When you cure and ferment tobacco, these tannins begin to break down slowly and are part of the unpleasant taste of young tobacco. The stuff is so strong that it was used for centuries in the tanning of hides into leather (hence "tanneries" and "tannenbaum" as the old German name for pine trees, a rich source of tannins). Anyway, these tannins are such long molecules that they take a very long time to fully break down. The longer the tobacco is aged, the more broken down they become, thus the aged cigar may seem smoother or rounder than a younger one. The breakdown of tannins also leads to pleasant flavor combinations that may not exist in a younger (or older) cigar. Since tobacco is a tannin intensive plant, we encounter tannins at many stages of decomposition. This is why we sometimes get fruity flavors from cigars, or chocolate and fruit, or cracked black pepper. The tannins in our cigars are similar enough to the tannins in other plants that we taste them as if they were those plants. Hope that's helpful.
