I don't think the thickness means much one way or the other, some SDs are as thin - I have one thin 512MB SD card and one thin 1GB SD card, amongst the other "normal" cards.
The biggest difference between MMC and SD is that MMC uses 1-bit transfers (i.e. it has 1 connector to transfer data) while SD has 4. But SD can also work in MMC (1-bit) mode (which has slower maximum transfer speed, obviously), and some SD controllers on some PDAs only uses 1-bit mode as well.
Now, MMC plus has extended to 8-bit mode, if I remember the article I read on wikipedia correctly.. certainly the one card I own is _stuffed_ with connector leads!