One of the most important problem is the question of logic and its relation, or lack thereof, to mathematical logic.
Once mathematicians started to develop their own method of logic, in the 19th century, mathematicians gradually came to be almost universally regarded as the experts on logic, probably to the chagrin of philosophers interested in or working on logic.
However, rather than the one method intended initially, different mathematicians started to develop, over time, different methods, with varying degrees of success and realisation. Some of these alternative methods resort from what mathematicians typically do, i.e. investigate whatever formal system they find interesting investigating, without necessarily any relevance to the real world.
Other methods, however, are clearly meant to be models specifically of the logic of human deductive reasoning. This was the case initially with George Boole and Frege's respective methods. This is also clearly the case with Relevance logic, at least in some respects.
For some methods, for example paraconsistent logics, whether they should be considered models of human logic will depend on who you ask.
It may be this situation that gradually led many mathematicians to view logic as essentially arbitrary. Any system will do as long as it is consistent according to its own strictures.
However, most philosophers today seem to fundamentally disagree with this perspective. Essentially, their interest, for most of them, is in the logic of human reasoning, not in whatever arbitrary construct mathematicians have come to call "logic". In particular, many if not most philosophers now don't take 1st order logic to be logic at all.
The result seems to be that during the last few decades there has been a renewed willingness among philosophers to look for some alternative model of human reasoning. Mostly, the idea is to assume that there is no deductive reasoning as such, rather that human reasoning is a mix of logics, such as deductive, inductive, abductive logic. I think they are wrong but this would be another discussion.
The result for the philosophy of mathematics is to shift the ground underneath the feet of mathematical logic. That is, the question now becomes that of the nature of the relation between mathematical logic and the logic of human deductive reasoning, and indeed whether there is any.
As I understand it, however, nobody for now seems to be willing or even interested in addressing this question. Philosophers seem to have lost interest in the topic of mathematical logic and are essentially looking elsewhere, leaving mathematicians to their own devices.
So, essentially, this is a question waiting for an answer. There is not doubt that it is important. While opening the question of human logic to the possibility of mathematical modelling, mathematical logic has very quickly lost its original focus on human reasoning, to the extent that mathematicians today don't have anything to say on it. This cannot possibly be without some serious consequences. Broadly, nobody is investigating deductive logic as such.
One difficulty is to characterise the work mathematicians are doing if not work on human logic. The question is so fuzzy and contentious, that most people, including mathematicians themselves, don't even understand it, or even understand that there is such a legitimate question to begin with.
The usefulness of an answer to this question would be in clearing the ground for a proper investigation of human logic. For example, in cognitive sciences, most papers on the subject of human deductive logic simply assume that a proper definition of it is provided by 1st order logic. And of course, they all conclude that no such a thing is to be found in the human brain.
So, an important question but a difficult one, perhaps, critically so because it may be difficult to understand the various issues without understanding logic first. A chicken and egg problem, which seems typical of logic.