If I may try my hand at an answer:
I don't think anyone with an iota of familiarity with algebraic topology has any serious doubts about the importance of the computation of the homotopy groups of spheres. What little is known about them is connected to very deep algebraic and topological information. For instance, as someone else commented on the related question, if one understood even the stable homotopy groups of spheres very well, one would therefore have a near complete understanding of the group (I assume that $n \neq 4$) of differentiable structures on the $n$-sphere: see e.g.
The tightness of the relationship between homotopy groups and differentiable structures recently took a big step forward via the work of Hill-Hopkins-Ravenel. (I was fortunate enough to hear a very nice expository talk on this by Prof. Michael Ching of UGA.)
I am already at the limits of my meager knowledge in this area, but I surmise that there would be other dramatic consequences of a systematic computation of homotopy groups of spheres.
The homotopy groups of spheres are also extremely challenging to compute. My understanding is that ever since Serre's work circa 1950, every generation of algebraic topologists has had a small number of luminaries who push the theory further -- modestly, but intriguingly. Therefore the combination of certified applicability to other results and intrinsic difficulty makes it easy to see why this has remained an irreresistible problem for algebraic topologists. In a roughly similar way, the Taniyama-Shimura conjecture was made irresistible by Ribet's proof in the late 80's that it implies Fermat's Last Theorem.
What is not clear to me (and again, I am an outsider in these matters to say the least) is to what extent anyone has ever been able to say, "Aha, I have computed sufficiently many homotopy groups of spheres in order to deduce the following striking consequence in [say] geometric topology". So far as I know, it may well be the case that the information flows in the other direction: one finds some link between homotopy groups of spheres and some fascinating and slightly more tractable other structure...and uses this to push a little further on the computation of homotopy groups of spheres. I.e., I wonder whether in practice, the homotopy groups of spheres are the applicationend, not the toolmeans.
In a perhaps too brief summary: if the homotopy groups of spheres were less complicated, they would probably be more directly useful but also less interesting.