Some things you did under the old Apple ID cannot be accomplished using the new Apple ID.
For instance, if you downloaded a game under the old Apple ID, you cannot update the game under the new Apple ID, because Apple's servers see the game and know that the new Apple ID isn't entitled to an upgrade for it. In a case like that you have three options:
1. You can provide the password for the old Apple ID so you can update the game, or
2. You can live with the game as it is, without being able to update it, or
3. You can delete the game and re-download it under the new Apple ID.
Just one of the prices you pay for abandoning the old Apple ID.