I think most people would agree that the criteria Gil Kalai highlighted are what we, as a community, would say to the layman, or to politicians, or other scientists (say biologists or chemists). But since we are on a math website, I am naïve enough to think we can be more honest.
In most cases, what makes a question important is that the big shots in your field decided that the question is important. This is perfectly illustrated by your feeling about Schur positivity. The big shots in representation theory decided that some questions are important and you feel that people are much more inclined to listen to you if you relate your interesting problem to the questions of the big shots.
But then you may ask "How to define a big shot". The answer is very simple : this is someone who solves important questions. And you get a perfectly virtuous circle of "important people" asking important questions and their students (or post-docs) solving them, becoming in the process the new big shots.
A long (long) time ago, I asked my master thesis adviser why is it that the students of such or such big shot would get permanent position so early in their career, while some other young researchers, who looked more productive, more original and more dedicated to Science in general, would have so much trouble to find a job. And he told me "The answer is very simple : the big shots get the best students to do a Phd with them, they give them the important and interesting questions for their theses and then the students get the best Phd theses, so they get the best positions early on in their career."
Edit added later : After afew more thoughts, I would like to argue that the word important should be avoided to describe a mathematical question. "Well-motivated, surprising, fun, well-connected,..." seem to me much more adequate to describe the mental reactions you may have when facing a (new) question.
"Important" is in my opinion a word of power that people with sociopathic behaviours in academia (see for instance this paper) may use to break the career of someone not working in their area ("this paper is not worth being published, it deals with unimportant questions") or to favour the careers of those closely working with them (" this Phd thesis is awsome, it deals with a very important question").