2 The comment was expanded, clarifications and an opinion was added.

So, my point is that the applications to physics mentioned above are misleading. Spheres which appear there admit lie group structure, and applications mentioned illustrate only the fact that Lie groups are tremendously important in physics. There is no single hope for spheres $S^n$ with $n \geq 10$ to have such sorts of applications in physics.

Of course, someone may cook up a "homotopical quantum field theory", where the key role will be played by spheres and their homotopy groups, but such a theory will hardly agree with experiment. (At least, at the current state of knowledge.)

In fact, I don't quite understand why someone could worry about "why the knowledge of homotopy groups of spheres is useful". I think the definition of homotopy groups is very natural (much more natural than many other definitions in topology which I have seen, at least to my (rather limited) understanding), and spheres are one of the simplest geometrical species. So, it is reasonable to try to compute those groups for the spheres.

The fact that we can't do this easily is already very important. It signifies that our computational power is rather (probably, shamely) limited. I don't think that this fact per se says anything about the importance of homotopy groups of spheres. They are important because they are natural things to ask and to compute. They also provide us with a challenge. There are many other things in math which are also not known.

Let me also mention, that as it is being repeated often, (here I refer to what is commonly said, am not a historian) the whole modern algebraic number theory appeared as a result of people's work attempting to solve Fermat Last Theorem. Could someone even ask "what are the applications of FLT in number theory"? Without FLT, there would be no algebraic number theory at all. (at least in the form we know it today)

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Sorry for posting this as an answer. This should actually be a comment (to the Paul's and Jose''s answers).

I doubt that there are (and will be in the nearest future at least) any applications to physics of higher homotophy groups of spheres, which are not Lie groups. Of course, something may come up accidentally, but this will hardly be a conceptual application.