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.
Yes - the "Power" tunic/robe/vest/sash (depending on the translation) is an item that is dropped by Behemoths. (King Behemoths drop it as well, but these only appear in the GBA version)
I think in the SNES version of the game it was just "(robe icon) Power"
The Final Fantasy Wikia gives its stats as:
It grants the wearer +15 to Strength and has 15 Defense. In the DS version it also gives 10 to Evasion.
I found a FAQ for the SNES version that lists the drop rate from Behemoths as 11/64 (see the comments below) - so you're likely going to have to farm some Behemoths if you're interested in getting one.
Best Answer
No. On the iOS version various things are missing. This includes the challenge dungeons. Other things that are missing are the extensive dungeon in the final chapter and Proto-Babil.