I'm working on a problem in $p$-group theory, namely "is it true that every finite non-simple $p$-group has a non-inner automorphism of order $p$?". (known to be posed or conjectured by Y. Berkovich)
My feeling about this question is being somewhat particular and isolated. I asked some specialists (who I'm really indebted to them) about this question, and their answers were very different: well known and quite difficult (a clever answer); important; particular and not very interesting; perhaps the methods that may be developed upon it, may have a great interest .
What make this isolated problem (with many many others) less interesting than Fermat's Last Theorem (conjecture) before its demonstration?
More generally, what make a problem in mathematics important or not?
Perhaps a problem is impotant if its solution implies the solution of other problems, but this amounts to saying that a set of problems is interesting, so we have the same problem.
And if we wish just to understand some things by solving a problem, then all the problems are interesting.
I'm sorry if the question is so vague.