I've been watching Bowhunt Or Die recently, and in one of the episodes they mentioned that you should give the deer enough time to "expire" after your shot, and not go looking for it immediately. When rifle hunting in the past, I've always gone looking for the animal immediately after the shot - is approach different with bowhunting simply because the animal would take longer to die from an arrow than from a bullet? When rifle-hunting, I've sometimes found the buck up to 100m away from its original location. How far can I realistically expect an animal to travel after a vitals shot with a bow?
Typically, about the same distance. I've seen them drop in less than twenty yards. But that's not the point of waiting. You want the deer to lie down and die. If you walk up on a live deer, he might kick you. If he does, you won't run far before you drop.
The old rule of thumb was to wait the length of time it takes to smoke a cigarette. When you don't pressure wounded animals, they are more likely to lie down and hump up. They won't travel as far, less adrenaline. It is a good idea.
The wait time for archery is to let the animal bleed out. The wait time for rifle is, as ppine said, gives them time to hump/stiffen so they have a hard time to get going again. Either way, its best to give them a few minutes. The key to approaching a downed animal is to approach from his rear or side and from uphill. Watch the eyes. If they are open and clear, he is not dead. If they are closed, he's not dead. A dead deer's eyes will be open and will glaze over immediately upon death. Don't laugh, there have been more than one hunter sitting astride his kill with the gun in the antlers when suddenly things got nasty. We found a badly abused model 70 with a broken sling one time. We had heard the story of its getting lost so it was great fun to return it to its highly embarrassed owner. I can't imagine the suffering that poor buck went through before the gun eventually broke free of his antlers.