Trackballs date back to the 60s, at least. The original design advantage of a trackball was pursuit and target acquisition. They were developed for military radar operators to quickly move the focus to an inbound bogie, potentially supersonic. If they could acquire and track the target on air search radar, they could mark it twice on two consecutive sweeps of the air search radar, giving the fire control system a course and speed so the fire control radar had a better shot of getting a lock.

My first four years out of college I spent a lot of time in that space, so I got quite comfortable with trackballs. Now I use them exclusively on my desktop. My only complaint is they aren't embedded flush in the desktop :)

The main draw is that you don't have to move your entire hand / the mouse along a surface.

That means your wrist sees less strain + motion, and you have no need for a mouse-pad, to name two possible benefits. Some people find it to be more precise as well, but I don't know if there's any evidence apart from personal preference.

Of course, your wrist still moves some as you manipulate the ball with your fingers, and if you use a trackpad you already don't need a mousepad-like-surface.

The trackball always does stay on the same place. You can have the trackball close to your keyboard, so you can click the "Enter" key from your numpad .. sometimes this is useful. I use mostly a trackball on work and at home.