I'm a Philosophy post-doc in the US. Here's my two cents on the issue.
Are there any subtler reasons to prefer one department over another?
. . . Does choice of department have any impact on postgraduate academic job prospects?
The primary reason to prefer one department to another would have to do with your plans post-graduation. If you want an academic job and your PhD is in philosophy, then it would be hard to be hired by a mathematics department. (I've seen it happen with philosophers of mathematics, but it's rare.) Since most professors end up teaching lots of introductory courses, which sounds more appealing: teaching lots of introductory calculus sections, or teaching lots of introductory philosophy? That's one useful way of thinking of it. If you want a non-academic career, the questions are a little different. My sense is that a math PhD is probably more "marketable" to industry than a philosophy PhD, but your mileage may vary.
Is there a significant difference in resources available to logicians in one department over another? (Perhaps philosophy departments tend to be more well-funded than mathematics departments, or perhaps there are more fellowship opportunities available to students in math departments.)
I would expect there to be more money for math than philosophy. Mathematics in the US is part of the prestigious STEM group of disciplines that attract lots of external funding. There is very little external money (comparatively) for research in the humanities.
Are there any significant differences in academic expectations? For instance, are there perhaps greater expectations to publish more frequently when in one department versus the other, or does perhaps one department take its language requirement more seriously than the other?
I don't think so. For a newly minted PhD in philosophy, you wouldn't necessarily be expected to have already published a lot. It's not uncommon for a new philosophy PhD to have one or two things already published or at least under review, however. My sense is that that is comparable with math, but perhaps somebody here can correct me.
Related to the previous question, are there any significant cultural differences between the two departments?
Philosophers can be very argumentative, and you'll be expected to know a whole range of stuff beyond just logic. There's a tendency among some philosophers to treat logic as basically trivia and not as properly philosophy. That's an unfortunate attitude. Likewise, I suspect that if you went to a math department, they'd probably regard the philosophical implications of logic as wooly-headed, poorly-defined problems that any reasonably confident mathematician could sort out in a long weekend, and that won't have any bearing on the real problems like . . .
Is it possible at all, once you are in one department as a graduate student, to switch to the other?
This would be administratively impossible at my university. You'd have to reapply to the different program, because of how funds for PhD students are apportioned by the university. There'd be nothing stopping you from taking a few courses at the other department, provided your chair and the prof whose class you were attending approved. But I think it'd be hard to do more than just a few classes.
. . . philosophy-related coursework has tended to focus more on continental philosophy . . . I have also heard rumors that philosophy graduate admissions tend to be, by virtue of self-selection and small entering class sizes, much more competitive than math graduate admissions . . .
Admissions to philosophy programs right now are really competitive. Let me also say that this is because of the terrible job market in academic philosophy. Prospective graduate students who want an academic career are advised only to apply to very selective programs in the US. Something like 50% of the tenure track jobs in philosophy the last few years have tended to go to the top ten ranked programs in the field. (Look at the Philosophical Gourmet Report for information on these.) Be very careful about attending any philosophy PhD program outside the very top! The result of this is that admissions at these programs is extremely competitive.
A bit of bad news though is that all of those programs are resolutely "analytic" programs rather than "continental" ones. There are good programs in the US where you might be able to find a faculty member or two with interests in continental figures, such as UChicago, UCBerkeley, and Columbia. If you're interested in admissions to one of these programs, then you'll need to make sure that at least one of your letter of recommendation writers is a famous enough philosopher to be well-known by the faculty of those institutions. There aren't a whole lot of "continental" philosophers who would be well known at those places, which may make your path a lot tougher.