Is there an example of a topological manifold in which different smooth structures give rise to tangent bundles which are not isomorphic as topological vector bundles?
My (our) own attempts or remarks, mostly obtained from discussing this question with other people:
By the Wu formula, the two tangent bundles would have in any case the same Stiefel--Whitney classes. This is in fact what originally motivated my question.
Exotic 7-spheres are no good to produce such examples because they all have trivial tangent bundles.
"Different smooth structures correspond to different (stable) linear structures on the tangent microbundle."