Completely different versions.
FFIV for the DS was basically a complete retooling of the graphics and added CGs and what not to it, but most notably in 3D.
PSP version is more of a mega boost of the original game, but still maintaining some of the feel and classical looks of the original game.
Also as others have mentioned it does come with after years as well, but the DS and the PSP versions will be greatly different so if you pick it up they won't really be the same experience because of how different they are.
Here's a small comparison in what I mean if you still don't understand,
Oh damn the PSP version doesn't have the garbage 3D models. I now regret not buying it.