Well...I guess "to me" refers to the movement of the ball. It means you want someone to take the ball from one location to your location. You can say it to your dog. For example, when you'r playing with it, although it will not understand for sure. You can also say "Bring me the ball" meaning the same.
When you use "for me" you sound like you are asking someone a favor. Like "Please, bring these balls for me, they're too heavy and I'm tired".