All the answers so far have focused on 3 dimensions, but the answer is much more striking in 4 dimensions.  Freedman's theorem tells you that classical homology invariants give you complete information about topological, simply-connected 4-manifolds.  These classical invariants cannot, however, distinguish between distinct smooth structures on the same topological 4-manifold, and essentially our only technique for distinguishing smooth 4-manifolds is Donaldson's invariant or the Seiberg-Witten invariant or their relatives.  These do not quite form a TQFT, but are related to TQFTs.