After defeating diablo the prime evil and then exiting a game by leaving instead of talking to the angel *it still does the end credits for me etc.. and unlocks the proper achievements for completing the difficulty* if you select to do a previous mission in a previous difficulty before resuming your game you have to do the quest over again before it lets you in to the new difficulty.
iirc (I don't have the game open atm) You can choose your difficulty in the same place as you choose which toon to play. There should be a drop down menu on the bottom left for it. Alternatively, if you join a co-op game, that can change your difficulty as well. Could one of these be the issue for you?