Late to this game, but the answers here seem to me to have changed the question.  Would it not be most faithful to read this as asking of the categories enriched by all the qualities of a module category are equivalent?

First is to agree on what one means by:

> Does this mean that the corresponding categories of finite dimensional complex representations are isomorphic (ignoring the forgetful functor to vector spaces), or just that the corresponding representation rings are?

I think for me the most direct interpretation is to take this to mean the category of $\mathbb{C}G$-modules. 

Take therefore the characterization of module categories by Morita (see [1,4.11].  Here one finds the to be a category of modules you need to be:
 * Abelian.
 * Gorthendieck (some intersection conditions)
 * Finitely pro-generated.

Every category of $R$-modules for a ring $R$ has those properties and furthermore, if $P$ is some pro-generator then define $S:=End(P)$ and you get a ring such that all the objects in this category become $S$-modules and the morphisms become $S$-linear and in fact the $S$-module category is equivalent to your given category.  This is what Morita says.

So the question thus becomes, when are two rings $R$ and $S$ going to give equivalent module categories despite perhaps not being isomorphic rings.  That is, more-or-less by definition, to say that the rings $R$ and $S$ are **Morita equivalent**.

So now we have a computable question.  When are two group algebras over $\mathbb{C}$ (or over any ring of coefficients) Morita equivalent?
Once we compute:

$\mathbb{C}D_8=\mathbb{C}\oplus \mathbb{C}\oplus \mathbb{C}\oplus \mathbb{C}\oplus M_2(\mathbb{C})$

$\mathbb{C}Q_8=\mathbb{C}\oplus \mathbb{C}\oplus \mathbb{C}\oplus \mathbb{C}\oplus M_2(\mathbb{C})$

These are not only Morita equivalent they are isomorphic rings.  So these two categories are in fact equivalent.

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I think many later distinctions are enriching the intended categories.  For example, one could say treat the rings $\mathbb{C}D_8$ and $\mathbb{C}Q_8$ as rings with involutions given by the inverse on elements (Hecke algebras).  Then they cease to be isomorphic and you see a difference.  Likewise other posts seem to enrich the categories with other qualities.  But to me those seem like exceptions to the most superficial interpretation of the question.  In any case, those who find confusion may benefit from teasing out the implications of the implicit assumptions.

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[1] Pareigis, B. (1970). Categories and Functors. Translated from the German. Pure and
Applied Mathematics, Vol. 39. New York: Academic Press.