Banjo-Kazooie was one of the defining games of the N64 era along with Super Mario 64. Banjo-Kazooie was able to take stuff that happened in SM64 and somehow made it better. So many more movement options with using the c-buttons to add to the traditional A and B button gameplay (fairly similar to Smash using the d-pad, mind you), not having to exit the level after collecting a jiggy allowing for more freedom and continuity in how you played, and Kazooie's flight was much more satisfying than Mario's because it wasn't timed, but resource based, giving me a freedom of movement I hadn't had in a game before and I feel like was the first game to truly realize 3D movement. Super Mario 64 was the first and most revolutionary, but I feel like Banjo & Kazooie, more than any other platformer, perfected it.
It's the king of the collect-a-thons, a genre of platformers, but they made the collecting satisfying as many collectibles aided in your abilities rather than simply being something to collect or aesthetics (something both Yooka Laylee and Mario Odyssey I felt failed to fully realize). It's a simple game, with nine worlds, ten jiggies and 100 music notes each, and the over world felt like a humongous level in itself. There were various platforming challenges and places that required out-of-the-box thinking, simply trying to get to the next stage, which is really satisfying. The worlds were very creative in taking traditional levels (e.g., "desert world", "snow world") but making them alive with unique and funny characters, often with their own subplots (and some truly dark ones). And the legendary music by Grant Kirkhope is rich in detail, whether it be changing the music as you went underwater or high in the air, to the transitions of the Gruntilda's Lair theme to match the style of each world. And while simple, it got incredibly hard, particularly in the 8th world where it got incredibly frustrating, but in a fun way that's often difficult for platformers to balance. Which made reaching the final world and boss all the more satisfying. I'd also be remiss to not mention the humor--from corny to punny, sarcastic to genuine, from childlike to dark, Banjo-Kazooie, and Rare in general, are known for their humor, which is fairly hard to replicate. Banjo & Kazooie make a hilarious duo, and Kazooie in particular gets some of the best sarcastic one-liners and meta humor. And particularly dirty humor, as Rare tried very hard to sneak in-jokes around Nintendo's heads, and I was surprised, even as a kid, what they got away with.
Banjo-Tooie was a worthy successor, and while it made the game more complicated in parts, the larger levels and harder challenges certainly made it engaging throughout. After Microsoft bought Rare, it became clear that they didn't give a **** about any of their old IP. A real Banjo-Threeie never came, despite being promised in Tooie, and instead what we got was a poor attempt at nostaliga grabbing with Nuts & Bolts while not understanding what made the two games truly brilliant. It's humor was criticizing the audience, rather than themselves, they completely changed the genre to be "hip with modern audiences" and the design of the characters lack any of the childlike charm the duo had in the N64 games. While Microsoft may own them, they will always be Nintendo characters to me.