As a 20+ year trackball user I swear by them. Gaming, graphic design, everyday computing, I love it. The perfect trackball has yet to be made but Gameball is pretty close. I used the logitech marble mouse for decades, probably went through 7 of them.
I've definitely found that I'm more accurate with a trackball. There's something about turning the ball to rotate the game camera that feels more natural to my muscles and to my visual perception compared to just dragging a mouse around, and my accuracy scores in games that track it (like Left 4 Dead) and online accuracy tests/practice back that up; I'm about 8% more accurate with a trackball than a mouse, on average. Obviously, this is going to vary by individual, so I'm not trying to make some broad proclamation that mouse users are wrong. But definitely, anybody who has used a mouse for years should give a trackball a quick try. They might be surprised.