Yep, I've got this one too. After an hour on the forums, it appears to be an iTunes 11.0.3 bug. I suspect that the address space randomisation in Mountan Lion is playing havoc with something that iTunes is supposed to remember.
The problem is NOT in iCloud (takes the new password every time), NOT in FaceTime (ditto), NOT in the Apple forums (ditto [whew!]) and NOT in IOS (iPhone/iPad both behave correctly).
If I type in the password as the very first activity before doing anything else with iTunes, it accepts the password. If I do anything else with iTunes before entering the password, my password is toast.