That is a abuse of power. The person that bans because of a personal grudge should not be allowed to continue to do so and should have that power taken away from them
While I agree in principle, the definition of a power abuse is a bit loose, since not everyone see it the same way, and it depends on whether you have the power or not.
When you're on a receiving end of the power stick, and you feel like you're being treated unjustly, you have every right to feel abused, but, as far as the person on the other side of the power stick is concerned, he (or she) is a god(dess) of his / her little world, and feels completely entitled and justified to hand out punishment to everyone under him / her as he / she sees fit, regardless of how anyone feels about it.
The only thing you can do in such situations is to either deal with it or - better yet - pack your e-bags & leave, since, in my experience, there's no much point in trying to talk to administrators, because most of them will simply play the rank card and dismiss you, even ban you if you don't take a hint and back off and let them win. 'Cause, if they ban you, it's still a victory for them because they get a satisfaction from an ego boost they just gave themselves by getting rid of you.
This is also what happens when the staff were all friends before the forum began, and operate as a clique to weed out those they don't like for whatever reason.
This.

I've seen it happen. Staff members vs regular members. If you don't fall into their clique, at best they will ignore you, or subtly provoke you until you give them enough material to get rid of you, making sure to make you look like a bad guy (or girl) and themselves as holy saints.