(Background: I'm a PhD student in logic, so I'm a little bias, but I've done a fair amount of work in both mathematics and philosophy.)
I'd probably argue that the philosophy-mathematics divide is narrower than most non-philosophers/non-mathematicians realize, but wide enough to be importantly different.
They certainly overlap, at least ideally, insofar as they both stress the use of systematic, critical thinking. Both put emphasis on arguments and on making distinctions to help elucidate problems. They also overlap in practice, insofar as they both tend to fall short of this ideal. Many mathematical proofs are given informally, invoking intuitions and leaving out a number of the details ("left as an exercise"); in a similar fashion, many philosophical arguments are not stated/stateable in premise-conclusion form, and so their analysis is not always as systematic as one would hope.
But their methodologies are importantly different. Philosophers, for instance, are very eager to argue over the foundations, and often debate about the most fundamental aspects of their field. Mathematicians, by contrast, work in a more cumulative fashion, often using the work of other mathematicians as a springboard for more elaborate proofs rather than questioning the foundations of their field. That's not to say mathematicians don't ever disagree, or that philosophers don't use the work of previous philosophers as springboards. But when mathematicians disagree, it's often presupposing there is an objective way of settling the matter (though not always). Furthermore, when philosophers build off of other philosophers' theories, it's usually involves modifying (sometimes very fundamental) aspects of the original theory.
With regards to whether a philosopher should learn some mathematics, or whether a mathematician should learn some philosophy, I would say almost unqualifiedly in both cases yes. They overlap enough that they should be aware of at least some of the basics in each field. I'd probably say that about most fields though: most fields can benefit from studying philosophy and mathematics. Whether philosophers have something in particular to gain from mathematics largely depends on what area of philosophy you're talking about, and even then it may be less about acquiring a certain methodology and more about the content of the mathematics in question (some branches of mathematics could, for instance, contain theorems that have relevance to certain philosophical positions). Similarly, mathematicians certainly have something to gain from philosophy, but studying Nietzsche is probably not directly going to make you a better mathematician (though it may make you a better person).