Well, just because a recipe tells you what your final gravity might be, doesn't mean that's actually what it will be. So don't use the number alone to know when your beer is done.
As for letting a beer continue to sit once it has finished fermenting, first the yeast go through a clean-up stage, where they eat some of the stranger flavors produced during fermentation. This is essential to making good beer.
After clean up, you're really in the "aging" phase. My personal belief is to keg my beer at this point. Even though it's not at its optimum flavor, I like to keg it so that I can taste the beer get closer to its perfection. Some people, however, have much more restraint than I do, and they like to age the beer for 2-3 weeks. I definitely agree that a beer needs at least 3 weeks from the day it was brewed to taste great. It might be good before then, but it gets better.
Since I'm guessing you're bottling, you have the extra choice to make: whereas I continue the bulk aging process in the keg, your beers might be susceptible to different aging rates once they are bottled. Still, because I'm an impatient guy, when I was bottling I bottled at the same "age" that I currently keg my beer. I'd rather they carbonate and be ready to drink (at that optimum 3 week age might I add) with the downside of missing some bulk aging.