The recipes always seem to encourage you to ferment for only 4-7 days, bottle and then drink quickly. I would guess it's because if it said on the box, "Ferment for three weeks, then bottle and wait three more weeks", then people might not buy it. But trust us on this- wait AT LEAST two weeks before bottle. After the hydrometer reading is stable, wait a bit more. There are reasons for this- first, after the bulk of fermentation is done, the yeast are still busy. They even digest their own waste products, when there is no more fermentable sugar for them to eat. This "cleans up" the beer and makes it taste better.
Also, sediment will fall out of the beer, leaving a cleaner beer. It won't harm you, but I'd rather have it fall out in the fermenter rather than in my bottles! The more crud left in the fermenter, the the less "stuff" in your bottles, and the beer will look better besides. A week old homebrew will be murky and cloudy. An older homebrew will look more like a commercial beer.
A little age makes the beer taste better, too. It'll be smoother, richer, with a better mouthfeel. It will blend in better, too. The harsh hops notes will smooth and mellow, while the malt flavors will meld. A three week old beer may be kind of drinkable, but an 8 week old one is excellent.
I start drinking quite a few of my ales at 5-6 weeks old, but the best pint is always the last!
