A central problem of interest in topology is the calculation of $\pi_n(S^m)$ and the classification of manifolds in general. In 1961 Kervaire constructed a manifold that does not admit a smooth structure.
What is the significance of smooth structures for computing $\pi_n(S^m)$? If there is no relation for homotopy groups of $S^m$, is there any information about $\pi_n (M^m)$ ($M^m$ an $m$-manifold) contained in the knowledge of whether $M^m$ does or does not admit a differentiable structure? What can be deduced about homotopy groups from the (non-)existence of smooth structures?
The wikipedia page on exotic spheres is relevant. (I am not in my office, so I don't have better references handy.) If one denotes the group of exotic spheres in dimension $n$ by $\Theta_n$, there is a certain quotient group $\Theta_n/\mathrm{bP}_{n+1}$ by spheres which bound parallelizable $(n+1)$-manifolds. Then there is an injective map $$\Theta_n/\mathrm{bP}_{n+1}\to \pi^s_n/\mathrm{im}(J)$$ where $\pi_n^S$ is the $n$th stable homotopy group of spheres and $J$ is the so-called $J$-homomorphism. By recent deep work on the Kervaire invariant problem, the above map is an isomorphism, except possibly when $n=126$, where it may have index $2$.
I have a feeling I may be summarizing what you already know, and maybe you are asking for the details of the map above?