I've read that the DS version of Final Fantasy IV is significantly more difficult than the newer PSP version. Why is this so? (I haven't played either game, and I have no experience with any previous version of this game.)
Final Fantasy 4 on the SNES originally came in two flavors - the one for Japan, and the one for the US. 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. I believe the fights were also quite a bit harder.
The US version (released as Final Fantasy 2 in the US) was made "easier" to appeal to the perceived not-as-hardcore US gamer. 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.
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. The DS version is closer to the original Japanese difficulty, while the PSP appears to take a page from the US SNES version.
The DS version was also a complete 3D remake of the game, with all-new 3D models and environments. It was a complete engine overhaul for the DS's limited 3D capabilities. I believe they also changed the script significantly, since the original English SNES version is widely acknowledged as having been very poorly translated.
The PSP version appears to retain the 16-bit visuals of the original game, which probably means it's closer to an emulated version of an updated SNES ROM than the DS version. This would also jive with a previous re-release of FF4, which was part of a compilation with Chrono Trigger called Final Fantasy Chronicles on the original PlayStation.
The PSP version also contains Final Fantasy 4: The After Years which was an episodic "sequel" released on the Wii in the US in 2009.
The history of Final Fantasy 4 and all of its various ports is quite a bit longer, and I've omitted details (such as the Easytype version and the GBA port, etc). If you're interested in reading more about it, I'd suggest the FF4 article on the Final Fantasy Wikia.
Better/higher difficulty (enemies higher stats and last longer than 2 turns, and they also counter a lot), script, balance, 2 bonus bosses, 3D/cinematic presentation.
Party switching, 2 bonus dungeons, 2D/classic presentation. You can basically complete it on auto-battle. The PSP version also includes Interlude (it's short) and The After Years (it's like it's own game), the mobile/WiiWare followup to FF4. It's more or less a port of the GBA version. Pickup the PSP version if you haven't played it.
Based on my own personal experience as I have played all versions including the original Final Fantasy IV on the SNES, the PSP version is much closer to it in terms of difficulty while the DS versions pushes WELL beyond that. Even the mist dragon in the DS version poses quite a challenge while it was still kind of a push-over in the Japanese SNES version.
Another dead giveaway that the PSP version was based off the JP Final Fantasy IV is that you get all the JP exclusive abilities (e.g. Dark Wave), you can't see any of the secret passageways whereas you could in the original SNES North American version (as part of "making things easier"), and Ethers aren't sold in any shops.
So yeah, pick your choice. 3D graphics, actual voice acting, and an incredibly high difficulty on the DS or improved 2D visuals and something that closely resembles the original Final Fantasy IV on the PSP (although like Final Fantasy II PSP port (as in the Japanese FF2 that was originally on the NES), the rate you get ambushed and back attacked is a LOT higher than before and your inventory space is actually VERY limited too). For an easy as pie experience, there's the GBA version which WAS dumbed down like the SNES NA port while maintaining some of the features of the original JP FF4.