It's not an extreme opinion to say that one prefers the Blue fight to Red.
In the first case, you have Blue, set up as your rival. He's always ahead of you by just a bit, even from the start, when he picks the starter that has advantage over yours (in RB). He's always at every town before you, he beats every trainer before you, and even though you beat him in every battle, he still holds that attitude over you, that 'you'll never surpass me' attitude. Even up to entering the league, he fights you to 'test' you, even though you've been on the same path, and should've been equal to him. His presence gave you an extra reason to beat the game - asde from being the champion, you had to beat Blue completely, and KO his cocky attitude.
Finally, after getting through 4 tough battles, you expect to have gotten there before him, but noooope. There he is, waiting to bust you down from the top, as reigning Pokemon champ. And when you do it to him first, it's awesome.
In GSC, dealing with Red...after the whole surprise of Kanto being on the right side of Johto, and going through the towns again, you hear about the events of 3 years back. The player isn't being told that Red is waiting somewhere for his next challenge with you, they're told things that the player may have done in RB, with references everywhere. Newly remixed music is awesome, new teams for each Gym Leader is cool, alright. Oh, Cinnabar is essentailly gone, so cool. This and that changed, so awesome. After beating the Kanto leaders, you are directed towards Mt. Silver, because it's a place for the 'strongest' players. After plunging through the dark, you come at the top of the mountain (it looked like a cave in GS though, lol), and someone is waiting in the dark with his back turned...hey, that sprite looks familiar...
PKMN Trainer RED wants to battle!
Oh wow, it's the guy who was my PC from RB! But with a team Ash might have!
Now, the Red fight in itself is cool. His battle is different from Blue's though, not necessarily completely better in every way. For one, his appearance is like a nice surprise. You don't expect him, but it's shocking when you do see him, like an awesome bomb to the face. Blue's fight is the culmination of an entire game...he was like the climax to your story. Gold's story was intertwined with Silver, who was sort of like Blue, but with a worse attitude. Gold's relationshp with Silver was more of an outside observer, watching as Silver was eventually cured of his habits concerning treatment of Pokemon, not necessarily by the player's hand, but by Lance's, and his own desire to be a better person.
Wa there something between Gold(you) and Red? No, since Red was part of the entire nostalgia package that was Kanto. Not the equivalent of the past 'you', who fought and beat Blue years back with the team you trained for an entire game, but the character himself, 3 years older, evoking an ideal Ash team, with Pikachu, Snorlax, three evolved starters, and Espeon, for some reason (it was better in HGSS when he had the Lapras, evoking that whole Silph Co. incident). Buuut, since it's the same character you used, the fight acted like the cap on Gen 1 + 2: beat the champ as Red three years ago...beat Red, who is the de facto champ, as Gold. So it's cool like that.
So, different reasons to like both.
I'd put the Blue fight over the Red in terms of story value, whatever that mean in these games. But I still like the Red fight almost as much.