This is answered in [Crowley, Diarmuid J.; Zvengrowski, Peter D, On the non-invariance of span and immersion co-dimension for manifolds, Arch. Math. (Brno) 44 (2008), no. 5, 353–365], see here.
Specifically, in each dimension $>8$ there is a closed PL manifold admitting two smooth structures whose tangent bundles are non-isomorphic. One tangent bundle is trivial and the other has nonzero second Pontryagin class. See remark 1.3.
Such examples do not exist in dimensions $\le 8$ by Corollary 2.6.