What do the difficulty levels change?

Simple mode removes a lot of attacks from bosses and sometimes even omits entire stages of a fight. However, you will not be able to play or progress to the final part of the game because on Simple you do not collect the souls from bosses. You need every boss soul to progress.

From what I can tell, it only changes the boss's attack patterns and attacks in general. Some of the harder sections from certain fights are removed as well on Simple mode, but I would just play on regular anyway. You're getting the full effect of the game that way, and ends up being far more satisfying once you finally beat the boss.