It might be worth explaining why you shouldn't expect $R(G)$ to tell you everything about a group.  $R(G)$ is naturally isomorphic to the ring of class functions $G \to \mathbb{C}$ (the functions constant on conjugacy classes) under pointwise addition and multiplication, and as such the information it contains is precisely <strike>the multiset of sizes of each</strike> the number of conjugacy classes of $G$.  That's it!  No other information.  (Note that $D_4$ and $Q$ both have <strike>conjugacy classes of sizes $1, 1, 2, 2, 2$</strike> five conjugacy classes.)  

In other words, the abstract structure of the representation ring actually gives you **less** information than the character table; the character table at least hands you a distinguished basis of $R(G)$.  Without this basis, $R(G)$ can't even tell you what tensor products of representations look like.