Among those who stopped at Gen II, they probably "outgrew" the franchise and didn't continue. If you're referring to fans who enjoyed later generations as well as earlier ones, but not III, then I have a few possibilities. Prepare for a post I probably spent way too much time on.
It might be because Hoenn was the first time they tried to separate the region from the previous ones.
GSC's Johto was a direct sequel to RGBY's Kanto, and even connected to it, so I suppose many fans were expecting whatever came next to continue that tradition. When Hoenn was unable to link to the previous games, lacked some of the Pokemon available in previous regions (until FrLg and Emerald filled in the blanks in the Dex), had Teams Aqua and Magma rather than Rocket (until the remakes), and otherwise just seemed disconnected from the previous generations, fans ended up with something they weren't expecting and may not have been sure how to react. Sinnoh had the benefit of Hoenn already having separated itself, so it was better accepted by following rather than starting a pattern. Unova had the benefit of two generations of precedent and being advertised as a "new beginning" from the get go. Additionally the 4th and 5th Generations allowed for Pokemon from previous generations to be transferred forward, unlike the gap between the 2nd and 3rd.
It may have also been that Team Aqua and Team Magma had goals that seemed a bit less logical than Team Rocket, even compared to later Teams.
Rather than seeking money and power, they wanted to better the environment by essentially destroying the environment. I'm guessing that a number of fans saw their plans as ultimately unworkable as expanding the land/sea for the benefit of people and Pokemon seemed to ignore the fact that it would expand the environments for some while shrinking that of others, which would in turn throw off food chains and cause other problems. Basically, it looked like Maxie and Archie didn't think things through. By comparison, Galactic's goal, while grander in scale, may have seemed more "reasonable" in that Cyrus simply didn't care about what happened to the old world in his quest to make a new one. While Galactic at least acknowledged that they would mess up the old world through their plans, Aqua and Magma seemed to think that they could make things better for their favored species without thinking of the consequences it could have for the rest of the environment, and even ultimately the habitats that they wanted to help. By Unova the goals shifted again from changing the physical world to changing the way humans and Pokemon would interact within it, so Plasma also seemed more reasonable than the Hoenn Teams.
As for the remakes of the 1st gen that were also a part of Gen III, it may have been because the fans felt they had already played those games before and didn't see the point of spending money on the updated graphics and Sevii Islands when they could still pull out their older systems and play R/B/Y.
Of the two, I would probably lean toward the fact that it was the first to be disconnected from previous generations that turned fans off the 3rd Gen. These are just my guesses, though. It could also be that they disliked the art styles changed between generations or other reasons (Pokemon losing its popularity and fans following the next trend instead, etc.).