iPod Touch Passcode doesn't work after restore from backup
The solution to my problem was rather simple, though I still don't understand it:

Reboot.

My battery drained completely, and when it came back on after plugging it in it presented the expected numeric keypad and accepted my passcode.  It seems rebooting was all I needed to get back to normal.

It doesn't make sense to my why it happened that way.  If you have an explanation, I'll happily accept your answer.