This is now the second time this has happened. I changed my Apple ID password on my other Apple device (iPad) and restarted my iPhone, then entered the new password on the phone, and it worked.
Last time, however, was a nightmare as all my password resets went to my iPhone, which wouldn't allow me in. Had to go to the Apple Store and talk to someone from their help line. She had one of the so-called geniuses switch the sim card in my phone with that of one of their extra phones, then reset the pw on my iPad, then reset it on their extra phone, and then restart my phone with the sim card back in place. Somewhere in all that, it got fixed. Tears and even a mall fire alarm later. This was incredibly frustrating.