Can exponentials be distinct from hom-functors in enriched categories?

Sure. Let $G$ be a non-trivial abelian group. Then the category of sets with a left $G$-action, $\mathbf{B} G$, is a symmetric monoidal closed category in two different ways. First, $\mathbf{B} G$ is a topos, so it is cartesian closed. The exponential in $\mathbf{B} G$ turns out to be essentially the same as in $\textbf{Set}$, in the sense that the forgetful functor $\mathbf{B} G \to \textbf{Set}$ preserves exponentials. In particular, the underlying set of the exponential $Y^X$ in $\mathbf{B} G$ is in general not the hom-set $\textrm{Hom}_G(X, Y)$!

Nonetheless, the hom-set $\textrm{Hom}_G(X, Y)$ does have a left $G$-action, namely the pointwise action inherited from $Y$. (This is where we use the fact that $G$ is abelian.) Moreover:

• The left $G$-action on $\textrm{Hom}_G(X, Y)$ is natural in $X$ and $Y$.
• $\textrm{Hom}_G(X, -)$ has a left adjoint, $- \otimes_G X$.
• $\otimes_G$ and $\textrm{Hom}_G$ together make $\mathbf{B} G$ into a symmetric monoidal closed category.