What is the difference betwen equivalence and isomorphism of functors in categories.
Well, the actual difference between the two statements is that for an equivalence of categories, we only require that that the composites $F \circ G$ and $G \circ F$ are naturally isomorphic to the identity functors rather than exactly equal. That is, there's a collection of isomorphisms $\eta_x :GF(x) \rightarrow x$ for each object of $A$ such that whenever $f: x \rightarrow y$ is a morphism in $A$, $\eta_y GF(f) = f \eta_x$, and a similar natural isomorphism for $F \circ G$.
As for why we do this... Imagine we're both doing group theory, so we both get ourselves a category of groups and start doing group theory in that category. But then we compare our categories and they're not the same: your category has one object for each isomorphism class of groups, while the objects of my category are given by a set $X$ along with a multiplication $\otimes: X \times X \rightarrow X$ which makes it a group.
Our categories aren't isomorphic, not by a long shot: for every object in your category there's a large class of objects in mine. So if we could only use isomorphisms of categories it would look like we're working on entirely different things.
Fortunately, our two categories are equivalent: using one functor which sends a set and a multiplication to its isomorphism class, and the other functor which takes each isomorphism class and picks a realisation of that group. Therefore, we're justified in calling both categories 'The Category of Groups' and any result you get in your category will also work in mine.