When introducing one's name, I have seen both of the following constructs used.
〜といいます。
〜というものです。
Is there a difference between them? I get the hunch that the latter is perhaps more formal than the former, but just how formal is it?
No difference. Both of them are polite or formal.
I'd say the latter is a bit more formal than the former just as you suspect, but not necessarily more "polite"; to me it sounds a little old-fashioned, though I'd take that impression with a grain of salt. In general, you wouldn't get in trouble for saying one instead of the other.
It should also be noted that you can also be even more polite by saying "~と申{もう}します (申{もう}す is the humble (kenjougo) form of いう, so it makes sense), though you may have already known that.
