The difference between the two versions, Leaf Green and Fire Red, are the obtainable Pokémon in each.
Just as with Red and Blue, each has certain Pokémon that you're unable to catch in a particular version. I believe they're the same as the originals for the most part, with the exception of the Gold and Silver Pokémon, as they've been added to the remakes.
Here is the list of which Pokémon are exclusive to whihc version:
Fire Red: Ekans, Arbok, Oddish, Gloom, Vileplume, Bellossom, Psyduck, Golduck, Growlithe, Arcanine, Shelldar, Cloyster, Electabuz, Elekid, Scyther, Scizor, Murkrow, Wooper, Quagsire, Skarmory, Qwilfish, and Delibird.
Leaf Green: Sandshrew, Sandslash, Vulpix, Ninetails, Belsprout, Weepinbel, Victrybel, Slowpoke, Slowbro, Slowking, Staryu, Starmie, Magmar, Magby, Pinsir, Azurill, Marill, Azumarill, Octillery, Misdreavus, Sneasel, Mantine, and Remoraid.
And because of this, I would advise you to pick the version which has the Pokémon you like.