It was a complete engine overhaul for the DS's limited 3D capabilities. The DS version was also a complete 3D remake of the game, with all-new 3D models and environments. The DS version is closer to the original Japanese difficulty, while the PSP appears to take a page from the US SNES version. Whenever Square Enix ports FF4, it's kind of random which version they opt to port, and they tend to make tweaks and other changes to the game as they go.
They also made a few changes to it, notably removing any references to religion ('Holy' is just called 'White') and removing any direct references to death.
The US version (released as Final Fantasy 2 in the US) was made 'easier' to appeal to the perceived not-as-hardcore US gamer. I believe the fights were also quite a bit harder. The Japanese version had several things about it that were more difficult, including the fact that you couldn't buy 'Ether' potions in shops that could restore your MP. Final Fantasy 4 on the SNES originally came in two flavors - the one for Japan, and the one for the US.
