Just like Red and Blue, the main differences are how often certain pokemon appear, which pokemon are exclusive to each version, and which color and pokemon appears on the "Press Start" screen (but they don't start randomly showing pokemon anymore after a few seconds).

In both versions, the story line is the same, the trainers are the same, and if you choose the same starter in either version, your rival has the same pokemon when you fight him every time. The wild legendaries (Suicune, Entei, Raikou) are the same assuming you choose the same starter. You only get 1 of these per version though (Bulbasaur=Entei, Charmander=Suicune, Squirtle=Raikou). And they still run just like in GSC... *Mewtwo CAN be caught with an Ultra Ball*

Mewtwo is still brooding in the Unknown Dungeon, which has been renamed Cerulean Cave and uses the Japanese map of RBY's Unknown Dungeon. Ho-oh and Lugia can be traded via the Gamecube Pokemons or through events like Mew, Celebi, and Deoxys (though the events have passed and you'll likely never see non-hacked versions of the 3 previosly mentioned pokemon).

Personally, though I dislike the color green, I recommend LeafGreen because it offers better pokemon like Magmar, Sandslash, Starmie, and the rare Sneasel.