Normal Mode: This is how you will play the game the first time. Everything is straightforward, and should match the walkthrough.
Replay Mode: Replay Mode is available after you finish the game once. You retain some BP and EXP from the previous game, and a few additional items are available in the shops, depending on your ending rank from Normal Mode. You can use BP to buy back PE activated in the previous game, but this is not worthwhile, as the amount of EXP required to activate these powers in Replay Mode is much less than in Normal Mode. Replay Mode is, in general, easier and more fun than Normal Mode.
Bounty Mode: Bounty Mode is available after your first game in Normal Mode. There are no additional items available for purchase, enemies are stronger, and more difficult enemies appear earlier in the game. PE powers cannot be bought back with BP in this mode, but more EXP is available due to the more powerful enemies.
Scavenger Mode: This mode could become available after your first game in Normal Mode, depending on your ending rank. In Scavenger Mode, there aren't many items available for purchase, and they cost too much. Aya does less damage and the enemies do more. If you play this mode, be ready to die a lot.
Nightmare Mode: You've got to be kidding. You start with low HP, strong enemies are present from the beginning of the game, and their attacks are very powerful. You will gain a lot more BP and EXP in this mode, though, if you can make it through it. Good luck.