I'm looking at translations of statements involving functions into the language of category theory, and it seems that there are at least two category-theoretic notions collecting functions between objects $A$ and $B$: exponential objects $B^A$, and $\mathrm{Hom}(A,B)$.
In $\mathbf{Set}$ these notions coincide, which leads me to wonder if the same happens in other categories. Since exponential objects remain in the same category, this seems only to be a sensible question in categories enriched over themselves. I know of two other such categories: $\mathbf{Cat}$, small categories, in which I believe the notions still coincide (although I haven't checked the details, I assume the notation $\mathcal D^\mathcal C$ for functor categories is no accident), and $\mathbf{Ab}$, abelian groups.
Now I'm a little stuck because hom-sets in $\mathbf{Ab}$ do have an abelian-group structure, but they aren't exponential objects, and indeed $\mathbf{Ab}$ has almost no exponentials. My guess is that this is essentially because the evaluation map $(f,x) \mapsto f(x)$ isn't a homomorphism of groups.
My question is: are there examples in which hom-objects and exponentials both exist, but are non-isomorphic objects of the same category?