More often than not, it's the "start from zero" bit. People were pissed off they couldn't get their precious Pokémon from GSC into this. Basically, they wanted another cakewalk of a game with the same old Pokémon every damn time. Having an all-new place with next to no links to all the old things was a no-go. I wanna see them deal better with a data issue as big as this one. I bet they'd rather not go through the hassle of making a pheriphery that'd allow the trade between Gen 2 and Gen 3 and make new games, which not only nets them way more money. It's ultimately better for the series as a whole.
Personally, I prefer the 3rd Generation even above the 4th. I don't know, I like it when the new Pokémon are, you know, actually new. 1st Gen did this for obvious reasons, and gladly, 3rd Gen also went this route, with only 2 Pokémon related to old ones. All the rest are 100% new, interesting and refresh the gameplay in a way GSC failed terribly to do. Training and getting to know new critters is where the true core of Pokémon lies at, and 3rd Gen did this really well thanks to the so-hated fresh start. In short, I'm glad it happened. I didn't want another damn Johto.
Hoenn was a much richer Region than Kanto and Johto, which felt samey. While this can be blamed on the system capabilities of the time, and even though Johto had amazing cities like Goldenrod and Ecruteak, as well as much more inspired locations than Kanto, Hoenn is the one Region that has an actual variety of environments all over it, and truly creative Cities and Towns. They're not all the same or generic, as they were in previous Generations, with a few exceptions. We've got a desert, a volcano, a vast sea, underwater areas, jungle, mountains, and a lot of lush and wonderful sceneries, which really stand out. Then we have Fortree, Sootopolis, Pacifidlog, Lavaridge, Fallarbor... even the more "generic" Towns and Cities have a charm of their own. Going through RSE was truly an experience thanks to the scenery alone.
The characters were all cool. I prefer the Johto Gym Leaders, but the Hoenn lot were also pretty cool, especially as far as looks are concerned. Also, RSE gave us a decent amount of supporting characters. Faces that stand out alonside the Gym Leaders and the Elite Four, which were few and counted in previous Generations. As for the evil Teams... the rivalry concept was really well done, at least in Emerald, where both stood as antagonists and their Leaders got differentiation. No matter how retarded or short-sighted their plans were, they had quite the spin on the tired Team Rocket stuff of the time, which was much-welcome. And personally, I like it when they stared to focus this one plot on the Legendary Pokémon. It's just a realistic take, whatever people say. Why should humans limit themselves to their own capabilities or methods, when they have an array of creatures to take advantage of in order to reach their objectives? Team Rocket did the same, but only Magma, Aqua, Cipher, Galactic, Go-Rock Squad and Dim Sun have actually taken this to the next level. And I like when such important Pokémon are actually involved in the plot, even if it forces one to catch them when it happens. It just adds more of a climax to the Evil Teams part.
The new features have enriched the gameplay and the core as much, if not more, than the Physical/Special spit, or the Special Stat split. Natures and IVs make each Pokémon truly unique, and Ablities are also interesting twists in the battle scene. The EV cap also adds to the strategy and the uniqueness, not to mention that it empowers a Trainer's style, which was merely limited to Pokémon and Movesets before it. Likewise, Pokémon feel more alive. No one can say that giving them Natures was detrimental. Even if some suck depending on the Pokémon, now they at least feel like true characters, and less like numbers, which was much more apparent in previous Generations, with only the Friendship to go by regarding a more "personal" attachment.
Pokémon Contests also added a new side-quest of sorts to focus on, not to mention a new Role in the Pokéworld and a way to make otherwise "useless" Pokémon in battles to have more of a sense of accomplishment. I dislike how they've been treated since Emerald, but it's a nice add-on nonetheless. Not to mention it adds more of a dynamic compared to the straightforward and non-interactive gameplay of the battles. And regarding the battle scene, we have the Battle Frontier, introduced in Emerald. Having more than one way to battle, and all those twists and turns... it was much refreshing, whether you like it or not. It's not as mindless as the generic battling, definitely.
Really, the only thing 3rd Gen lacked was a proper use of Real Time. However, if it was going to be like GSC, or even worse, like DPP, where it really doesn't makes much of a difference... I'm glad it's not in. Thanks to that, they polished the game where it's actually important.
Regarding other games... FRLG was a nice remake overall, but they left out several needed bits which make it not too good. Not to mention useless "Help" features, especially the flashback crap. That aside, they were a huge improvement in every way over the original games. Colosseum and XD, no matter how many people hate them, gave a new dimension to Pokémon, expanded the plot, and also allowed you to actually focus on training Pokémon you otherwise would ignore. All of these games were great jewels in their own way.
I've been into Pokémon since 1st Gen, but I can say without remorse that 3rd Gen has been the most enjoyable for me. I'll never understand why people hate it. Having Pokémon you dislike doesn't makes it any worse than other Generations. There's nothing wrong with preferences either, but it gets ridiculous when random people call 3rd Gen "skippable". Funnily enough, I still play my 3rd Gen games more than the 4th Gen ones. That might change once I get HeartGold, but 3rd Gen will still remain as an outstanding Generation that wasn't scared of moving away from tradition and gave those capable of appreciating its greatness a much-needed fresh start in this series. Hell, hadn't it been for Gen 3, I'd have abandoned Pokémon by now.