Turaev defined a simple-homotopy invariant which is a complete invariant of homeomorphism type (originally assuming geometrization).
Here is the Springer link if you have a subscription: Towards the topological classification of geometric 3-manifolds
He claims in the paper that a map between closed 3-manifolds is a homotopy equivalence if and only if it is a simple homotopy equivalence, but he says that the proof of this result will appear in a later paper. I'm not sure if this has appeared though (I haven't searched through his later papers on torsion, and there's no MathScinet link).