I've been playing Stadium 2 on and off since it was first released, but having picked up a copy of Stadium a few weeks ago, I have taken a preference to the original's battling. You get a lot of close-ups on the Pokemon and I prefer the commentator's lines, such as "It's a One-Hit Wonder!" when you OHKO something in the face.
Well, when it comes to better (if only slightly) animation and graphics, and possibly the Gym Leader Castle, Stadium 2 takes the cake.
However, we only rented the game once, and I don't remember it very well. The boys and I still love the first one, it's just a lot of fun, and still manages to hold up to this day. We don't play it as often as we'd like (that's more for Pokémon Puzzle League), but if we wanted to do a tournament and "double" battles, now that we have the four needed, we can go to that game and just have fun. We always make sure the opposing player goes into another room when we're choosing Pokémon though, lol.
So because the memories are a lot fresher with the first, that's what I'm going for. I don't doubt the second one was really good, but I can't make a proper judgment on it until I actually get around to playing it all the way through.