One of the important things about letting meat chill or hang, even if all you can do is put whole quarters on the shelves in your refridgerator, is that that time allows the meat to go through rigor mortis, which makes the meat shrink up. I've heard if you cut up meat before it has a chance to go through rigor mortis, it's won't be as good. I've never really done it any other way, so I don't know what a deer tastes like that hasn't gone through rigor mortis at least while in whole quarters, sometimes all still one body.
I also have a family recipe that will make even the toughest rutting buck tender and tasty and unique from any other deer meat you've ever had, but my dad might ring my neck if I let his cooking secret out.