To bore you with more formal terms: All manipulations of a Rubik's cube form a group. And so do all the manipulations possible on a 2x2x2 Rubik's cube. And the 2x2x2 cube's group is a subgroup of the 3x3x3 cube's group, and there's a trivial homomorphism from the 3's group to the 2's group. That homomorphism is not an isomorphism.
I was aware of the group structure of the Rubik's cube (it is, after all, a subgroup of the permutation group on 54 elements), but apparently I had never given it much thought beyond that. Thanks!