can anyone fill me in on the advantages of a trackball over a mouse?
Trackballs do not care much about what surface they sit on, as long as it isn't something they can fall off of.
You don't move a trackball, so they can fit into very small space.
In a game, there is a 180 or 360 turning advantage (less so, when the trackball is super behind in dpi).
Trackballs often provide ergonomic benefits, reducing RSI problems, over mice.
A super tiny trackball, might generally be better than a super tiny mouse, but a trackpoint beats them both.
Trackballs, especially because the technology hasn't advanced as fast, also have some disadvantages.
The most obvious being that they are technologically behind mice, while costing more.
Another issue in the world of optical and laser, is that even though trackballs are now optical or laser, they still require more cleaning than a mouse, where one used to be able to get away with cleaning them less.
Scrolling on trackballs is often different. Depending on the trackball and your personal tastes, this can be worse or better than mice.