A funny/alarming side note on the subject: I've changed my password on Twitter, but I haven't had to enter the new password on either OS X Mountain Lion or my iPhone!
I'm not sure how Twitter works, but it's generally considered to be a bad idea to store the password on the device after you login (because someone could then look at the password file).
Instead of doing that, they check the password when you login and then download a token unique to that device and account allowing the App to connect to the service.
Some services handle revoking these tokens differently. With something like Dropbox, you can "Unlink" a device (lost/stolen/selling it/etc.) without changing the password. Other services might do it automatically when you change the password.