I'm probably gonna get hate for not having this generation higher on the list (especially in these forums) but eh. This is where I feel the generation belongs.
First, let's get the negative out of the way. Next to Black and White, Ruby and Sapphire had the absolute worst. story. ever. It all rests with the evil teams, Aqua and Magma. Both their plots were just outright stupid. Yeah, yeah, eco-terrorists and all, but the thing is that eco-terrorists usually have some level of logic behind their plans. Archie and Maxie lacked the most basic level of common sense. I'd say their followers were functionally retarded, but that would be a disservice to people who are truly disabled. Not even Emerald could fix this issue (but we'll get to that in a minute.) At least Team Plasma had an interesting concept that was butchered in execution. Teams Aqua and Magma have a terrible concept made even worse with their execution. Being that Ruby/Sapphire were my first foray into Generation III, I can't say I was impressed.
But let's talk about all the good things Generation III did. There's a lot to plow through, so bear with me. Let's start with Emerald, which took every positive thing introduced in RS and expanded on it, all the while dropping a lot of the stupidity along the way. Sure, Teams Aqua and Magma were still utter idiots, but at the very least, Emerald seemed to focus less on their stupid goals and more on the turf war going on between the two, which was nice. Also, when Kyogre, Groudon, and Rayquaza were awakened, I have never been happier to see a spotlight stealing squad. Their cutscene together was absolutely gorgeous, and it was nice to have to deal with two equal but opposing GODS
instead of the petty and idiotic evil teams.
Next, let's talk about FireRed and LeafGreen. The remakes of Generation I could not have come at a better time. With these remakes, Game Freak managed to fix just about every single problem that the original games have, bar the lack of story (also, still no canon Missingno.) They updated the franchise's flagship game to the newest generation at the time, and the games looked gorgeous. Their mechanical issues were fixed, and they also allowed us to use some of our old favorite Pokemon (Bulbasaur!!!!) In addition, they expanded the post game by, y'know, actually having one. This post-game, however, ended with a whimper instead of a bang, which was disappointing. It was nice to have some shoutouts to the Johto arc though.
Now, I must include Colosseum and XD. While not technically canon to the main series, these two games for the Game Cube were truly amazing (and they're canon in my head, so shut up.) Not only were they visually stunning (I recently replayed the games and they look almost as gorgeous as some Wii and 3DS titles,) they introduced something I had longed since given up on: a darker, more mature storyline. The post-apocalyptic(-ish) setting was beautifully rendered and the anti-hero Wes was such a badass. XD managed to have a much improved story at the cost of their badass protagonist. But it's alright, Michael's existence is forgiven because Shadow Lugia. It also let me use Johto Pokemon, which were otherwise scarce in Generation III.
Finally, the mechanics. Aside from introducing abilities, natures and streamlining the IV system, Generation III built upon many things introduced in Gold/Silver. These include, but are not limited to: an improved berry system, battle frontier, breeding mechanics, and weather. Aesthetically, Hoenn is damn beautiful and I wish I could see it with XY graphics. The Pokemon introduced were also among the best batch in Pokemon history (personal favorites include Sceptile, Blaziken, Manectric, Milotic, Gardevoir, Swellow and Absol, among others.)
Sorry for the length of this one, but there was a lot to cover in Generation III.