When you complete the game any new game playthroughs will be harder. The current difficulty level is represented by crystals on the Main screen. Every game clear adds one crystal and when you have 10 crystals it becomes a large crystal. The maximum you can get is 5 large crystals which is the equivalent of the difficulty being increased 50 times.
At the last 'Save Point' save the game to unlock the Seraphic Gate on the world map. This secret dungeon has new characters and items and when you have beat it and re-entered it again the enemies will have become more powerful.