The same topological manifold given two distinct differentiable structures comprises two different smooth manifolds. Do these two smooth manifolds have the same Stiefel-Whitney numbers? In other words, are the two differentiable structures cobordant?
If not, what about the case where one of the smooth manifolds is a boundary?
Wu's Theorem states that for a closed manifold $M$, $w = \operatorname{Sq}(\nu)$ where $w$ is the total Stiefel-Whitney class, $\operatorname{Sq}$ is the total Steenrod square, and $\nu$ is the total Wu class; see Theorem $11.14$ of Characteristic Classes by Milnor and Stasheff. The expression $\operatorname{Sq}(\nu)$ only depends on $H^*(M; \mathbb{Z}_2)$ as a module over the Steenrod algebra. Therefore, given two closed homotopy equivalent manifolds, we see that they have the same Stiefel-Whitney classes, and hence, the same Stiefel-Whitney numbers, so they are cobordant.