John, I can guess, but, never having seen drywall screws made or read a process routine for them, I'm not sure. I believe drywall screws are strong since they are probably heat treated. The problem is that the same heat treatment that makes them strong also makes them brittle. I'm guessing that they are quenched and not tempered or tempered well. I would also bet that the tensile strength of the drywall screws is higher than that of nails by quite a bit but there is a lot less area in one than you think there is and the notch effect from the thread, combimed with the brittleness makes then snap.
However, the solution is easy. Next time you are at the home center or plumbing supply store, buy a toilet wax ring for about $1.00. Don't get one with a plastic collar, just the plain cheap one. Leave it in the plastic wrap and stick the screws into it. The wax lubricates the screw going in and you will break a lot less. Leaving the wax in the plastic makes it less messy, for awhile anyway. Try this and I think you will be pleasantly surprised. Let me know how it turns out for you.