If $V$ and $W$ are both finite then clearly $\operatorname{Dim} (\operatorname{Hom}(V, W)) = \operatorname{Dim}(V)\cdot\operatorname{Dim(W)} = \operatorname{Dim}(\operatorname{Hom}(W, V))$ so they are isomorphic.
I'm not so sure if one is infinite. An "infinite matrix" construction for a linear transformation from $V$ to $W$ would have $\operatorname{Dim}(V)$ rows each with a finite number of non-zero entries, while from $W$ to $V$ it would have $\operatorname{Dim}(W)$ rows each with a finite number of non-zero entries. This suggests that perhaps $\operatorname{Dim} (\operatorname{Hom}(V, W))\ne \operatorname{Dim}(\operatorname{Hom}(W, V))$ ?