I don't have a Sony system, but I do own an iPhone. From the App Store, I have downloaded and I am currently playing FFT-WotL. From what I can tell it's the same as the original, with a few tweaks for it to work on a touch-based device.
There aren't two versions within the Apple App Store. I'm curious, what are the differences between the two PSN versions?
Language & Naming. Many Character names are spelt a bit differently and the language has changed a bit (said to be more medieval w/e that means). Many skills and skillsets are changed in name. e.g. The Knight's skillset Break has changed to Ruin. Speed Break -> Speed Ruin.
Special Characters. Luso from FFTA2 and Balthier from FF12 are obtainable party members.
Jobs. The Dark Knight is an obtainable job (This is not to be confused with the original's Dark Knight, Gafgarion which is now called Fell Knight). The Onion Knight is not allowed any abilities (class & secondary) but is able to equip any equipment including the famous Onion Knight equipment. More on the Dark Knight, Fell Knight, and Onion Knight on their wikia pages.
Side Quests. Additional side quests involving Agrais, Alicia, Lavian (Agrias' Knights), Beowulf, & Reis. In no particular order. I believe there are only two.
Multiplayer. Multiplayer was added to War of the Lions.
Delita - There are a few battles in the game where you're able to control Delita as a character that weren't in the original as well (to avoid minor spoilers I'm not mentioning when).
Main source (among the others listed and my own knowledge). I felt the need to post as the original post didn't mention any specifics and left it all to a link, which will eventually die making the answer not as good, so I added the info in another answer to maintain the reliability of the question's answer(s).
War of the Lions is basically the PSP version/remake, released 2007. It features a number of improvements and new elements (which are described on Wiki page in more detail - watch out for story spoilers though!) over its predecessor. Notable changes include increased difficulty level (reflecting the difficulty of original 1997 Japanese version), as well as additional battles, characters from later FF games and completely new translation. The iPhone version is a port from PSP, with only slight changes.