Consider a problem like Fermat's Last Theorem. I have no clue what it's about, and probably wouldn't begin to understand it if it were explained to me. But obviously it was a seductive story; the proof Fermat said he'd worked out, but that had somehow been lost. Were subsequent attempts affected by changes in the field of mathematics over the next 350 years? That would be, from the point of view of someone like me, an interesting idea to pursue. And perhaps easier to deal with than arriving at the proof itself, though not by much. So count me out. But as far as I know, solving that kind of mathematical problem is no more "useful" than determining the year in which Leonardo da Vinci's portrait of Cecilia Gallerani was painted.
I did part of my PhD with a group that had been extravagantly funded (relatively speaking) in part to prove Fermat's Last Theorem using mathematics built on a funky new logic.
Didn't work - big sadness!
For me, the unanswered question was: Why care about FLT anyway? (Apart from the glory aspect.)
You can see why mathematicians cared about it, though - very simple statement, looks like it "should" be easy to prove. It finally fell to very complex mathematics, certainly way beyond whatever Fermat had available to him.