The difficulty setting is a bit of a red herring. The only major changes are that the enemies have more Hp and Sora takes more damage from attacks. The A.I. doesn't get any better or anything. As for the story side of things. All that is affected is the conditions for getting the secret ending.

In Proud mode you just need to finish the game and you get the ending. For the other difficulty modes, you need to complete Jiminey's Journal of doom. This means basically unlocking everything and achieving the high scores stated in the Journal. You need to beat the game's optional boss, complete the Synthesis section, and complete with the high scores all of the Hades Paradox Cup.

If you are just playing for story, I would choose Proud mode and play through that as the difficulty isn't really that great cause Sora is a god with this games combat system. If you don't care, choose any of them and complete the request at your leisure.