Take any two closed connected homeomorphic smooth closed 4-manifolds that are not diffeomorphic. Then their products with $\mathbb R$ are diffeomorphic because the smooth structure on a 5-manifold is unique. It follows that the original manifolds are tangentially homotopy equivalent, i.e. there is a homotopy equivalence that pulls stable tangent bundles to each other. A priori this homotopy equivalence need not be homotopic to a homeomorphism but if one of your manifold is stably parallelizable, so is the other one, and then the homeomorphism has to preserve the stable tangent bundle because the pullback of a trivial bundle is trivial.