@RoboKaren's advice is good for many fields, and the basic point that you should know something about the departments you're applying to certainly applies in mathematics — you don't want to, for example, say it's your dream to study number theory in an application to a department which has no number theorists. However, mathematics is extremely specialized, and incoming PhD students (in the US, at least) are not expected to have a very specific idea of what they will work on. Most likely, you wouldn't even understand the abstract of your PhD thesis if you were to read it now. This is true even for the exceptional few students who have done truly substantial undergraduate research; typically their PhD research is only very loosely related to it.
The implication of this for which schools you should apply to is that you should generally aim for departments which are strong in a broad spectrum of fields. Fortunately, there are lots of them.
It's good to say something about your research interests in general terms, but the admission committee understands that by the time you start on your thesis they may have changed radically (say, from group theory to mathematical biology). If you yourself clearly don't understand that, it can create the impression that you don't know some basic things about how the field works. (And of course you don't; you haven't even been to graduate school yet! But nevertheless, you don't want to draw attention to that ignorance unnecessarily.) Bringing up research interests is more important for demonstrating that you've been exposed to some substantial mathematics than for indicating why you're interested in a particular department.
So, to finally come to your actual question:
I'm applying to math graduate programs at the moment. How important is it to tailor the personal statement to the different departments to which I'm applying?
Unless you've already discussed possible thesis topics with someone in the department — which is extremely rare in mathematics and not at all expected — I would say, not much. Mentioning broad categories like "geometric analysis" is sufficient — again, as long as there actually are geometric analysts on the faculty!
All of this is intended as general advice for "typical" prospective PhD students in mathematics in the US (where "typical" includes even most successful applicants to top programs). The exceptional few who have substantial research as undergraduates and want to continue in the same direction presumably have research mentors who can give them advice tailored to their personal situations.