After you first complete Lightning Returns: Final Fantasy XIII, you’ll unlock the “Hard” difficulty, game design director Yuji Abe and battle design director Nobuyuki Matsuoka revealed in a Famitsu interview.
Hard difficulty monsters will be stronger and drop higher-level abilities. There will also be items that are only sold in Hard mode.
Additionally, the number of each monster type living in the world of Lightning Returns is predetermined. When you defeat the last of their kind (properly known as the “Last One”), the monster will become extinct. Each Last One will be remarkably stronger than their counterparts, and depending on the monster, their behavior may differ as well. If you defeat a Last One, you might acquire rare items.