Yeah, I guess the title pretty much speaks for itself. While the Sonic Adventure games are my favorite games of all time, I freely admit that there are plenty of flaws that could be ironed out or improved to make a truly fantastic 3D Sonic game. In that way, Sonic 06 succeeded in some regards, and despite the game's infamous reputation I think it's only fair to point out the things it did right. I'm not saying it's a better game than Adventure 1 or 2, not by a long shot, but I do genuinely believe it did several things better than its predecessors. With that said, I'll get us started.
1. Making Sonic feel like the main character of the game
A common complaint I hear about 06's story is that Sonic himself has a miniscule role in the plot, and all of the important events are covered in Shadow and Silver's stories, but I don't agree with that at all. I'd certainly agree with the Adventure games, since he had absolutely no connection to Chaos, Tikal, Professor Gerald, the ARK, etc. Rather than being a genuine character with a narrative role in the story, Sonic was more like a plot device to fight the bad guys and save the day simply because he was the only one strong enough to do so. In 06, however, that's not quite the case. True, for most of the story Sonic's only goal is to save Elise from Eggman, and remains oblivious to the bigger picture with Iblis, Mephiles and Solaris, which are covered in Shadow and Silver's stories. However, I don't think it's fair to say that Sonic has no role in the story because of that. Shadow and Silver certainly have their own important agendas, but ultimately they're secondary to the real meat of the story, which is Sonic's relationship with Elise. At the beginning of the game, Sonic admits he has no special reason to want to help her, but over the course of the game they develop a strong bond, which eventually blossoms into romance. Mephiles wants Sonic killed so that Elise will cry and release Iblis, Silver goes back in time to do just that in hopes it will save the future, and Shadow needs to prevent Silver from doing that in order to expose and defeat Mephiles. Ultimately, all narrative threads lead back to Sonic. While Shadow and Silver resolve their respective arcs at the end of their stories, Sonic's is not resolved until the last story, where his death at the hands of Mephiles sets everything into motion. After Solaris is defeated and Elise refuses to blow out the flame, it's the bond she spent the game developing with Sonic that gives her the strength to do so. Now, it's perfectly fair to argue that Sonic's story was the least interesting, or the most poorly-written of the three, and even that it's not fitting in a Sonic game to begin with, but to say that he had no role in 06's story is simply not true.
2. Level design that compliments every character
This is something that Adventure 1 was sporadic about. Since the game's 11 levels were shared between 6 characters, overlap was inevitable. In the cases of Knuckles, Amy, Gamma and Big, they did for the most part have altered level design and layouts that catered to their unique abilities. However, Tails was by far the worst offender. In his case, he was utterly broken because he simply had to go through stages that were clearly designed for Sonic, not him, and could effortlessly fly through every obstacle Sonic would normally encounter. Even worse, Tails' levels were littered with dash rings that could help keep him in the air even longer. The only level that tried to give Tails unique level design was Speed Highway, and even that was a breeze to fly through. Now, Adventure 2 did away with this problem entirely by having each character go through stages unique to only them. And while that led to more balanced level design, I think it took away the charm of having characters go through the same places, but in different ways. In that way, I think 06 struck a nice balance. All 3 hedgehogs go through the same 9 stages, but in different orders and in ways that compliment their individual playstyles. In Dusty Desert, for example, Sonic runs through the desert with Elise, Shadow rides through the desert on a hovercraft before entering the hidden ruins, and Silver travels through a temple with puzzles that require his powers to solve. This also extends to the extra characters that could momentarily take over for the main trio. When Tails/Knuckles took over for Sonic, the level design changed to compliment flying/gliding. When Blaze took over for Silver, the level became more speed-oriented, and so on. Because of that, I never felt like going through the same levels with different characters was repetitive. If the developers can create levels that can be experienced by different characters with different abilities and compliment each one in its own way, that's good design in my book.
3. Unique boss fights
The last point sort of bleeds into this one, but I still think its worth mentioning on its own. In Adventure 1, the boss fights were almost an afterthought, either because each character had so few of them or the ones that they did have were incredibly samey, in both story and gameplay. With Chaos 4, for example, he was fought by either Sonic, Tails or Knuckles depending on whose story you play, all at the exact same point in the story and defeated in the exact same way. Adventure 2 wasn't much better, since it was mostly either character battles or G.U.N. robots that weren't much different from each other. While 06 does have some offenders of this, like the Egg Cerberus and Iblis Phase 2, for the most part Sonic, Shadow and Silver not only had a decent number of bosses each, but the ones they shared were fought under different circumstances and defeated in different ways. Sonic fights the Egg Genesis in Soleanna Forest, and defeats it by attacking the head. Silver fights it in White Acropolis, and defeats it by deflecting missiles back at it. Another thing the 06 bosses do that the Adventure bosses (mostly) didn't was take advantage of the unique terrain. When Sonic fights the Egg Wyvern, he crashes it into debris falling from the Egg Carrier, while the base itself is firing down on you. During Silver's first fight against Iblis in Crisis City, Iblis picks up entire skyscrapers and tosses them at you, the rubble of which you can fling back at him. I'm not necessarily saying the bosses themselves were especially well-designed, but the attempt at making each encounter more creative and immersive was appreciated.
That should give us a decent starting point. If anyone else has any additions to the list or has anything to say about the examples I posted, feel free to discuss!