Difference between a limit and accumulation point?

The difference is very simple.

1) As you wrote: an accumulation point of a set is a point, every neighborhood of which contains infinitely many points of the set.

2) But a limit point is a special accumulation point. No matter how small neighborhood you choose, all members $a_n$ (after a certain $n$) are in the neighborhood of the limit point.

The requirement for all members (after certain $n$) is obviously stronger than the requirement for infinitely many points/members.

So every limit point is an accumulation point, but not every accumulation point is a limit point. Also note that: (1) if a sequence has a limit point, then that's the only accumulation point of the sequence; (2) if a sequence has more than one accumulation points, that this sequence has no limit point. Try to prove these two, it will clear your confusions.

Unfortunately, there doesn't seem to be one standard definition for theses terms. One author may say that accumulation points are limit points, others do not. Best thing to do is to check the definition given in book, paper, etc that you are reading.