A few comments on applications that aren't covered by the above Wikipedia article.
I don't know any applications to cryptography. Most cryptosystems require some kind of one-way lossless function and it's not clear how to do that with the complexity of the homotopy groups of spheres. Moreover, the homotopy-groups of spheres have a lot of redundancy, there are many patterns.
If I was to start though I'd probably try relating
There's work by Fred Cohen, Jie Wu and John Berrick's work where they relate Brunnian braid groups to the homotopy-groups of the 2-sphere. Regardless of It's not clear if that having has any cryptosystem potential but it's an interesting aspect of how the homotopy-groups of a sphere appear in a natural way in what might otherwise appear to be a completely disjoint subject.
Homotopy groups of spheres and orthogonal groups appear in a natural way in Haefliger's work on the group structure (group operation given by connect sum) on the isotopy-classes of smooth embeddings $S^j \to S^n$. I suppose that shouldn't be seen as a surprise though. Moreover, it's not clear to me that this is always the most efficient way of computing these groups. But I think all techniques that I know of ultimately would require some input in the form of computations of some relatively simple homotopy groups of spheres.
I think one of the most natural applications of homotopy groups of spheres, Stiefel manifolds and orthogonal groups would be obstruction-theoretic constructions. Things like Whitney classes, Stiefel-Whitney classes and general obstructions to sections of bundles. Not so much the construction of the individual classes, more just the understanding of the general method.