The

If I'm not mistaken, the extra information is not contained in the Hopf algebra structurerepresentation ring, you have to look at the category of representations. In particular, you want to look at the representation category equipped with its forgetful functor to vector spaces. Then the group can be recovered as the group of group-like elements (if $\Delta$ denotes the coproduct, then the group-like elements are those which satisfy $\Delta(x) = x \otimes x$). Just considering the algebra structure (over the complex numbers), you only see the dimensions automorphisms of the irreducible representations (by the Artin-Wedderburn's theorem http://en.wikipedia.org/wiki/Artin%E2%80%93Wedderburn_theorem)this functor. Here's a related questionblogpost I wrote which may be helpful: http://mathoverflow.net/questions/500/finite-groups-with-the-same-character-tablehttp://concretenonsense.wordpress.com/2009/05/16/tannaka%E2%80%93krein-duality/

The extra information is in the Hopf algebra structure. In particular, the group can be recovered as the group of group-like elements (if $\Delta$ denotes the coproduct, then the group-like elements are those which satisfy $\Delta(x) = x \otimes x$). Just considering the algebra structure (over the complex numbers), you only see the dimensions of the irreducible representations (by the Artin-Wedderburn's theorem http://en.wikipedia.org/wiki/Artin%E2%80%93Wedderburn_theorem).

Here's a related question: http://mathoverflow.net/questions/500/finite-groups-with-the-same-character-table