Ha ha, I finally found it. Under Preferences -> General Preferences -> Playlist there is an Advanced Playback Settings section. Under that there is a Manual playlist advance checkbox and it was checked, meaning I had to manually advance the songs. Of course, it doesn't say if there is a shortcut key, so I have no idea what I hit that changed this setting.

Also, it didn't help that Winamp has about 50 different screens under Preferences and I'd been through them a gazillion times already...