Final Fantasy IV

Not much really except that certain things get carried over like augments equipped on people or the ones that you havent equipped. The only items that get carried over are the special items like the apples and the summon items as well as the tails for the onion equipment. And what ever you filled up in the dungeon maps will remain the same although from what heard as well is the if you didnt complete the Hummingway side-quest than you cant do it in the second or third playthrough and as a result, you wont get the augments that he provides. The big difference in the second playthrough is the ability to fight two optional bosses that you can only fight if you had stolen the item from the final boss. Aside from that, it really is more use for stat-maxing

I was just wondering, are enemies the same levels on the next playthroughs or do they get tougher.

All agments carry over along with rare items, like tails, onion gear, unuses agments, and Rydia's rare drop summons, soma drops, silver, and golden apples, and dark matter if you stole from Zeromus at the last battle, and all the agments you gave to all the characters also stay on them for the next game. You just have to equip them when you get that character.

darkragnarok pretty much has it covered, save for two things:

1. Stealing Dark Matter from Zeromus is only necessary for fighting one of the two optional bosses. The other one is available in New Game+ whether or not you have Dark Matter.
2. The Namingway quest is not unable to be completed in New Game+, it continues from where you left off in it. If you'd gotten as far as Puddingway but couldn't find Rainbow Pudding, he'd still be Puddingway in New Game+, waiting for you to find and bring him the rare and elusive Rainbow Pudding.

Also, I don't believe he mentioned the things that carry over, but Deidon covered that.

The biggest difference is the appearance of two strong, optional bosses that you can meet along the way.

Also, at the end of each game, the scene of Kain on Mt. Ordeals shows his face a little more each time.

Aside from that, you just get to keep the augments and maps that you had in the previous game.