Mainly it's graphical differences, the Sevii Islands, and available Pokémon.
The Gen III games have updated graphics, being able to take advantage of a more powerful graphics processor. Also, there are are several new additions to FireRed, including the Running Shoes, the Vs. Seeker, and the (worthless, IMO) Fame Checker. Also different is the Pokémon movepools. Most Pokémon in this game can learn moves introduced in Gens II-III, as well as most of their old moves from Gen I. The Sevii Islands are a small subquest that you first take part in after beating Blaine, as well as after you beat the Champion for the first time. After you beat the E4 for the first time, they allow you to catch Gen II Pokémon. Also unique to these games is that they allow you to connect to Ruby/Sapphire/Emerald, which Red/Blue cannot do. The plot between the two games, however, remains largely the same.
Between FireRed and LeafGreen, however, there are virtually no differences. The only major difference is version-exclusives. Check Bulbapedia for those differences.