I run GBA so I'm bound to miss a few things, but from what I understand:

PS1 is basically a cleaned up reskin of the NES version (includes bug fixes & still uses Spell Charges instead of MP). There are some additional options such as "Easy" mode which reduces shop prices, gives more stats per level and reduces EXP requirements per level, and the option to toggle whether or not attacks input on enemies that were defeated redirect to still living enemies.

GBA/PSP doesn't have a difficulty option and attacks on dead enemies always redirect to still living enemies. Uses the MP system for magic. They are pretty much to Easy mode from PS1. Adds a dash feature for faster dungeon/town movement. Adds modern items such as Phoenix Downs. There's also added bonus dungeons (4 in GBA, and PSP gets those 4 + 1 additional new dungeon).

PSP uniquely has a glitch which allows for equipment to be equipped to the wrong slot, as well as change what that equipment is (ex. the game pulls the wrong memory index and thinks a Hammer is the Knight's Armor. You can then put that Knight Armor on the Glove Slot and Helmet Slot on top of the normal Armor slot). Glitchless PSP plays largely the same as GBA i believe