Nostalgia will hit many fans of Red and Blue (or Green in Japan)
The difference in both versions is which Pokémon you'll be able to capture. For example, in FireRed, you'll be able to catch Ekans and Psyduck whilst LeafGreen has Sandshrew and Bellsprout. The only possible way to completely fill up your Pokédex is to find someone else with the opposite version of yours and trade with them. Traded Pokémon can grow faster. However, if you don't have enough gym badges, they may not listen to you.
There are many trainers willing to battle, and if they spot you, a "!" appears over their heads and they rush to you for a fight.
The game supports the now-probably-useless GBA Wireless Adapter, a little device packaged with both games. By plugging it in on the front of the GBA (sorry, DS users. It doesn't do it for you.), you'll be able to enter the union room at the Pokémon Center where you can chat, battle or trade wirelessly. Of course, if your buddy doesn't have an adapter but can go wire-style, you can still trade with the GBA link. There are some mini-games you can play as well, such as the Berry Crushing mini-game, the Kanto equivalent to the PokéBlock creation in Hoenn.
All in all, many Red and Blue fans will see some key differences in the GBA versions (like the TM Case and VS. Seeker), but the game still has what the GB versions use (and sometimes lack...so to speak). The strange difference is that some people will find trainers that have Pokémon that weren't available in the GB version. As in, you can get a Bulbasaur from Professor Oak, but later you'll fight a trainer who has a Squirtle. The tradition is carried over in Diamond and Pearl, mainly so the player has a chance to fill up their Dex.
If you really enjoyed the GB versions and want to experience a remake, you'd better pick either of these two games up (and hopefully, if you have the money, get BOTH of them instead of one). There aren't many fans of the games as I am of them, so you'll have to look REALLY hard around your town for people who still like these games.