Caroline comes to New Orleans to see Klaus because she is without her humanity and craving some extra vampy fun. She left Stefan in Mystic Falls because Stefans default setting when people he knows flip the switch is to get all 'memory lane' and 'telling them over and over that they have to allow emotion eventually. Caroline doesn't want to do that right now. She wants to forget and Stefan is part of those memories.
Klaus not being tied up in a relationship since Hayley married into the werewolf pack is ripe for a Caroline sighting.