Before I say anything, I'm not calling adventure a bad game. I just think it was a wrong turn made by SEGA that led Sonic on the path he is on today.
A big problem for Sonic today is he doesn't have a strong identity like before and I believe adventure is where it started, but then proceeded to influence games that came after.
Story
Sonic Pre-Adventure:
Before Adventure, Sonic would always start in a green nature themed stage with bright colors. This may seem insignificant but it gave us an idea of where Sonic came from and as you run through the stage you would destroy badniks that clearly looked out of place in contrast to the leafy areas you came across. It was clear what you are fighting for.
Sonic Adventure:
The first thing we see is Sonic at station square. We see humans going about their business but then.. Sonic comes along. All of a sudden its SONIC who looks out of place. His bright colors are at odds with the realistic setting and textures which he is in. We now have the police, the military and the government the tone doesn't fit and the plot takes itself too seriously. This is the first sign that Sonic Team have lost his identity.
Robotnik Pre-Adventure:
The first appearances of eggman come in the form of bosses. But these bosses were more than just machines. They always had a bit of meaning to them. They showcased what Eggman represented, which is industry. So in all Sonic was about nature(Sonic) vs industry(Robotnik).
For example we had:
Hei Hou - Who wielded an axe and chopped trees down
Fire Breath - Flamed trees down with the flamethrower
Drilling Robot - Which destroyed the natural beauty of Mystic Cave or the ancient ruins of Marble Zone
Mega Mack - Which polluted the sea with chemicals from chemical plant
Tails Pre-Adventure:
People like to complain that Tails isn't what he used to be. That they should do make him cool again like in Adventure. But actually the adventure games did just as much damage to Tails as all the other games that came after.
Before we had Tails follow Sonic through his chase for Eggman. But he always stood out when he brought out the tornado. The tornado was one of those 'rusty ol reliable' kind of planes. But what would Tails do? He'd fly that thing up to the WING FORTRESS. When we see the scene in Sonic 2 as he flies across it, we are supposed to think of the courage required to do that and we know Tails is the underdog.
Tails Adventure:
In Adventure we see the tornado now has the ability to shoot and take down enemies, and then Tails makes it transform into an even faster jet. It gets improved in SA2 when we see Tails in a walking mech that shoots lasers. Tails has suddenly lost his underdog status. But even if you believe he hasn't, it still shows Sonic Team forgot what made Tails special. Which is why we see wild variations in his portrayal today.
Gameplay
Multiple Characters
Prior to the adventures when a game had multiple characters it limited them to 3 with minor variations to the story. And anytime a character had a full fledged story to themselves it would be a spin-off (Knuckles Chaotix, Tails Adventure). This made it easier to keep development within reasonable costs. But SA1/2 while I applaud their ambition, it also had the effect that now many fans DEMAND that any Sonic game with multiple playable characters they must also come with their own campaigns (I'm ok with it but I know many aren't). Unfortunately today that would blow up development costs especially with current and next gen, and it seems SEGA decided to abandon the concept altogether. And to be honest they can't go back to what they did because the brand has lost too much power and non-Sonic characters can't sell a game on their own anymore.
Gimmicks and alternate playstyles:
It's ok to like the different playstyles offered in SA2 but it doesn't change the fact that they were not what the series was about. I would LOVE it if they forced me to play breakout before every session of Sonic but I know that isn't the reason people play Sonic games. Unfortunately since adventure this has become some kind of staple of the franchise with Sonic 06(Telekinesis) Unleashed(werehog), Colors(wisp), Lost World(∞).
Level Design:
In the classics, the level design did not separate speed sections with platforming sections. It was all one integrated whole, which allowed the game to flow smoothly at every skill level. For SA1 I could forgive them as it was their first (proper)outing with 3D. But SA2 didn't give much to hope for. The level design would clearly show you that this is either a platforming section or speed section.
The main reason for this was Sonic Team disregarded a main aspect of Sonic's identity which was momentum based gameplay. When they saw a shuttle loop they thought it was a 'spectacle' when in fact its an obstacle! They'd completely missed the point and so decided to place dash pads right in front.
The homing attack was also introduced. And it would've been a great addition to Sonic's arsenal (maybe even necessary in 3D) had the level design not had been built all around it. Contrary to popular opinion, the homing attack can in fact live in harmony alongside pinball mechanics. This is because you lose all momentum when you home in in an enemy as compared to bouncing off. And with clever level design it can advantageous to bounce off enemies instead of homing to reach new areas. But there have been far too few cases to convince me that Sonic Team understands this.
With all said, I must admit I'm big 'boost' fan and I know the style wouldn't have existed had it not been for Adventure. But even I can see the path they have taken may not have been the best.