John Milnor gives an example of two homeomorphic smooth manifolds whose tangent bundles are not isomorphic as topological vector bundles, see his ICM-1962 address, Corollary 1. I think, this was the first such example.
Edit. Few more things (motivated by questions in comments below):
- The total space of the tangent bundle to any exotic $n$-sphere is diffeomorphic to $TS^n$, see
R. De Sapio, Disc and sphere bundles over homotopy spheres, Math. Z. 107 (1968) 232-236.
- If $M_1, M_2$ are two homeomorphic manifolds, then the total spaces of their tangent bundles $TM_1, TM_2$ are always homeomorphic (the tangent bundles are even topologically isomorphic as microbundles), this follows from
J.Milnor, Microbundles-I, Topology, 3 (1964) 53-80.
- I do not know of any examples where tangent bundles of two smooth homeomorphic manifolds $M_1, M_2$ such that the total spaces of $TM_1, TM_2$ are not diffeomorphic, but I did not spend much time thinking about this either.