One knows that the tangent bundle of an exotic sphere is bundle isomorphic to the tangent bundle of the standard sphere.
Are there closed manifolds that admit several different differentiable structures in which two of the tangent bundles are not bundle isomorphic?
In Corollary 1 of this paper, Milnor constructs two smooth manifolds $M_1, M_2$ that are homeomorphic but have $TM_1$ trivial and $TM_2$ not even stably trivial. The proof involves constructing a bundle that's stably trivial ($s$-trivial in the notation of the paper) in the category of microbudles but not stably trivial in the nicer category of vector bundles.