I never thought that just mentioning PBR in this thread would create such as big discussion. I never owned any of the Stadium games and I only played (at least one of) them very little a few times while visiting a friend, so I can't speak for how PBR was compared to them. I always wanted these games as a kid, but I never even had a N64 and by the time I could get a console and the games, they were already a thing of the past. Which was why I got a Gamecube and Colosseum/XD instead.
Either way, I wanted to continue a bit with the discussion of PBR, so here are some specific things I liked about it (some have already been mentioned by others):
- All of the different colosseums with their different rules and restrictions, offering many sorts of challenges.
- The music. As said, I loved the Pokémon selection theme. I also liked the Minor boss battle theme, Sunny Park Colosseum, Sashay's theme, Magma Colosseum, Sunset Colosseum, Stargazer Colosseum, and probably others too.
Here's a playlist with all the music from the game if anyone wants to listen.
- Character customization! PBR was way ahead of the main series games with this feature. Another interesting thing about it was how you could sort of choose the age for your character, IIRC they had three different "models" for both genders, with some being notably older. You could also decide on what your trainer should say before and after battle, which has never been possible in the main series games at all.
- Seeing all of my Pokémon from the Gen 4 games in full 3D on my TV, that was really cool. I also thought some of the graphics in PBR were impressive at the time, it looked better than Colosseum and XD in several aspects.
- Some of the challenges were really fun and hard too. I specifically remember the last rounds of Stargazer Colosseum being pretty tough, and winning 100 battles in a row at Courtyard Colosseum with semi-limited healing was very challenging too. Granted, most of the Pokémon I used did not have perfect IVs (this was long before I learned Gen 4 RNG) nor were my strategies always the best, so that might have made it a bit harder than it should have been otherwise.
- Being able to get various items from the game, notably some that couldn't be farmed in the main series games. I remember that Choice Specs could only be obtained once in each Gen 4 game, but you could get an infinite amount of them from PBR. Then there were TMs and a bunch of other useful items as well.
- Getting some exclusive Pokémon from the games as well. There was a Pikachu with Surf and Volt Tackle that you could get after beating Mysterial for the first time. It was also possible to get a Electivire/Magmortar if you inserted a specific code, which was cool (though I also remember that Europe had to wait longer than other regions as usual before we got our codes).
- While I never used PBR for this myself, it also had online functionality and random online battles which I believe wasn't available in the main series games from Gen 4? So that's good, but also bad in a way, I guess.
But PBR isn't perfect. It has some issues, others have stated several of them already. For me personally, I think the biggest issue was the lack of a "story mode" similar to what Colosseum/XD had, and the connectivity between PBR and the main series games for the DS was a bit wacky in some ways too. But even so, I really liked PBR for what it was and I had a ton of fun with it back during the early Gen 4 days.