North America * Final Fantasy IV * Izuna 2: The Unemployed Ninja Returns * The Mummy: Tomb of the Dragon Emperor * Nancy Drew: The Mystery of the Clue Bender Society * New International Track & Field
Europe * King of Clubs * Lifesigns: Hospital Affairs * Pony Luv * Road to Vegas * Yamaha Supercross * Buzz! Master Quiz * Go! Explore * International Athletics
Japan * Kangaeru Ryoku o Gungun Nobasu! DS Youji no Nou Tre * Katekyoo Hitman Reborn! DS Flame Rumble Hyper - Moeyo Mirai * Kikansha Thomas DS 2: Asonde Manabu DS Youchien * Kirarin * Revolution: Minna de Odorou Furi Furi Debut! * Koma Neko DS * Korg DS-10 * Kung Fu Panda * Kuwagata Tsumami: Kuttsuke! Tsumami Hako * Machi-Ing Maker DS * Moetan DS * New International Hyper Sports DS * Nippon Futsal League Kounin: Minna no Futsal * Speed Racer * Touch de Taikyaku! Yawaraka Sensha * Twilight Syndrome: Kinjiratera Toshi Densetsu
Australia * Fashion Dogz 2 * Imagine Rock Star * Imagine: Girls' Band * Sid Meier's Civilization Revolution * Suite Life of Zack and Cody: Circle of Spies
I assume that you know the SNES version, I'll use that as the base to work from.
The GBA version adds abilities removed from the North American SNES version, such as Dark Knight Cecil's "Darkness" attack, Palom and Porom's Bluff and Cry, Yang's Focus, Tellah's Recall, and Rosa's Pray. It also gives you the option at the end of the game to customize your party if you choose, allowing you to take any (surviving) character to the final dungeon rather than the SNES version's unchangable five. Two bonus dungeons are also included, one to get new equipment for the characters you can now take to the final dungeon (since they didn't have late-game equipment before, as they were unplayable at that point) and another unlocked after the game is defeated, giving extra story insight into each character and a chance to unlock ultimate equipment, before fighting an ultimate boss.
The DS version doesn't have the end-game character swapping or the bonus dungeons, though it does have a couple of bonus bosses. It has a redone 3D game engine and voice acting in some cutscenes. The abilities that were absent from the SNES version return, as in the GBA version. Some monsters have been rebalanced, usually to offer more challenge. The Sand Worm by Kaipo near the start of the game is a real threat now, and gives XP to match that threat. The game also features "Augments", which can be found while playing or given to you in plot events. Augments let you attach new abilities to characters, often abilities of characters that leave the party, but some with entirely new abilities as well. Sample Augments include the Counter augment which lets the equipped character counterattack physical attacks, and the Phoenix augment that resurrects other characters when the equipped character dies at the cost of all of his MP. Augments are single-use and cannot be unequipped. However, the game has a New Game+ mode when you complete it, and your Augments (equipped and unequipped) and very rare items (rare summons/Adamant Armour/new Onion Equipment) carry over from game to game. You can thus reacquire Augments in your new game and give them to different characters, to try to form the ultimate party and challenge the bonus bosses.
Also, FF4 Advance (American version) has a nasty bug during battle where characters go out of order, get their turns skipped, etc. The slowdown can get pretty grating. I think you can change the battle settings so you don't see the ATB bar.. and I highly recommend doing that. Otherwise, play the European version if you can.
FF4 DS is nastier in terms of challenge. Bosses do things now you don't normally expect, or they do the things they used to do but HARDER. When they tell you to defend against the Mom Bomb boss on Mt. Hobs.. do yourself a favor and defend.
FF4 DS also changed some abilities. Edward's Salve ability now gives you the full benefit of medicine for everyone. In the earlier versions, it would split up the effect (so a Hi Potion would do 100 recovery to everyone.. now it does 500 to everyone). If you give him the Item Lore Augment, it does double recovery, too! However, Cecil's Darkness ability got changed from the powerful Dark Wave to a nerfed ability called Darkness where your attack power increases but HP gets taken off every time you attack. I never used it, and he loses it soon enough anyways.
Otherwise, Chumpstain summed it up great. FF4A is good for the bonus stuff, and FF4 DS is the ultimate version of the game.