I used to think what's the point, you throw a ball and catch a ball who cares what it weighs, don't mess w/ it. I also used to feel like swinging a heavy bat a few times before I went up to the plate helped me swing a little better. Although it may have psychologically made the bat feel a little lighter and more comfortable in my hands (increasing my ever expanding ego at the time) we now know that it did little to nothing for my actual hand and bat speed.
We know this w/ training as well. Taking BP w/ a heavy bat isn't going to help bat speed. And really I think it's probably going to mess w/ your mechanics a little bit and/or throw off your timing when you get in the game.
On the other hand, research shows that throwing w/ balls slightly above and below the standard weight of a baseball can improve arm speed. Weird. I made fun of those weighted ball sets when the first showed up in catalogs years ago.
Little side note for overbearing parents, not every kid needs to be using weighted balls. For some pitchers it's useful once in awhile but only when it's necessary. And it's not something you do everyday, nor is it something you start doing w/ everyone of your little leaguers. Be smart. These are people you're dealing w/ and you can screw them up way easier than you can help them.
Weighted balls have their place, but wont replace a solid strength and conditioning program or proper mechanics. Just one more tool.