Being isomorphic as representations of a group G

Let $G$ be a finite group. What is meant by two finite dimensional vector spaces (over $\mathbb{C}$) $V$ and $W$ being "isomorphic as representations of $G$"? To show that we have such an isomorphism, wouldn't it suffice to just show that $\dim V = \dim W$?

No, it doesn't suffice that the representation spaces have equal dimension. What is usually meant by isomorphism of representations, is that there exists an equivariant isomorphism between the representation spaces, i.e. one that also preserves the action of $G$. In other words, two representations $\pi: G \to GL(V)$ and $\pi': G \to GL(W)$ are isomorphic if there exists an isomorphism $A: V \to W$ such that $A(\pi(g)v) = \pi'(g) Av$.