In English, the Stadium series has 2 games: the original one compatible with RBY, and Stadium 2 which was compatible with GSC (though you could also use RBY games with it if you wanted to).
Meanwhile, the Japanese Stadium series is made up of 3 games.
The original "Pokemon Stadium" in Japan is more of a tech demo. It's compatible with all of Red, Green, Blue, and Yellow, and you can you can use the Pokedex function to view all the 3D models that you've seen in your connected game, but you can't use all of them in battle--there are only 40 Pokemon that have been approved for use there. There are also only two battle formats (plus the free battles you can take part in using multiplayer): the four-tiered "Level 1-30" division, equivalent to Nintendo Cup 98, and the single-tiered "Level 50-55" division, equivalent to Nintendo Cup 97 or Poke Cup. This latter tournament at least has a neat idea behind it: back in the day, there was a real-life tournament series around these formats, and 15 players made it to the final stage of the NC97 event which was held at Nintendo's own Space World expo. The trainers you face in that mode are based on the real-life competitors from that event, with their teams intact; the game picks 8 random trainers each time so you're not facing a consistent lineup like you might be used to from other Stadium cups. This game was also responsible for providing Surfing Pikachu to Japanese players: if you play the Master Ball of the level 1-30 cup with a Pikachu on your team, and send Pikachu into the final battle and win, it gets to learn Surf. Note that this game does not feature continues for winning battles in a 3-0 sweep like later offerings did, so this is a tougher obstacle than it seems. All in all, this game is probably not worth getting unless you're also getting the other two to complete the series.
The game called "Pokemon Stadium 2" in Japanese is equivalent to the original Stadium in English: by now they've made it so all 151 Pokemon can be used in battle, and added things like Gym Leader Castle and Round 2 modes to give more content to the game. The most notable difference is that while English Stadium has 4 different cups to choose from, this game has 6: Yellow Cup (aka Pika Cup), Fancy Cup (aka Petit Cup), Ultra Cup (aka Prime Cup), Nintendo Cup 97 (aka Poke Cup, but here it only has 1 division instead of the 4 it has in English), and two tournaments that we never got an equivalent to in English: Nintendo Cup 98 and 99. The 98 cup is the same as in the previous game, but now it's a one-division cup, and the Round 2 version of it does the same trick that NC97 did in the previous game: the opponents you face there are pulled from the real-life finalists of the NC98 circuit (and those real-life tournaments were played out using the original J-Stadium game). NC99 is a new four-division cup that kicks off the concept of "Underused" formats: all the Pokemon that performed well during the previous two years' tournaments, like Zapdos, Tauros, and Starmie, were banned from this format, so you had to look for lesser-used gems like Persian and Slowbro. Some of the rental Pokemon you can use in the various formats also have different moves than they do in English, but most players won't really care about this (unless they're wondering why there are almost no Fire-type moves available on the rentals in Fancy Cup). This is the game where English players got to unlock Surfing Pikachu for themselves, but that function isn't available in the Japanese version, since they already got it in the previous game. The other gifts (a random "choice" Pokemon for beating Gym Leader Castle, Psyduck with Amnesia for finishing the Hall of Fame) still work the same in both versions. And lastly, English Stadium got a Pokemon Snap-style mode where you could take pictures of your Pokemon, in exchange for not having those two extra cups; there's no picture mode in Japanese.
Finally, you would expect the Japanese game based on our Stadium 2 to be called "Stadium 3", but it's not; instead they called it "Pokemon Stadium Gold-Silver" (金銀). This has all the same formats as in English, and no extras; there are a few minor moveset differences but you would really have to be scouring for them to notice (and most of them are in Earl's Pokemon Academy). The English version of this game has a uniquely colored cartridge, where the frontplate is a glittery gold color and the backplate is a glittery silver; not so in Japanese, where it simply uses the standard gray N64 cartridge.
If you do go ahead with getting Japanese versions, there are a couple things worth noting: while these games are electrically compatible with English N64s, they are not physically compatible by default, because of a plastic "region key" installed in the cartridge slot. There are a couple ways around this problem; one is to unscrew the backplate of the game and replace it with the backplate from an English game, possibly a cheap throwaway sports game, and then it'll plug in and work just fine. The other option is to open up the N64 system itself, and unscrew the plastic piece from around the cartridge slot, which will allow you to play both regions' games without having to swap around the backplate for every game. Additionally, Stadium games won't accept cross-region GB games in the Transfer Pak, so if you have any of those games in English, you won't be able to import your Pokemon to use with them. Luckily, the Japanese GB games also tend to be cheaper, if you want to import Pokemon from those instead and not have to be limited to rentals.
