How about for starters the first level. The return of Green Hill Zone. Classic and Modern flavors.
The glorious return of the song E.G.G.M.A.N. in the Mirror cutscene, and then again outside the Boss gate for Egg Dragoon.
Classic Sonic gets one over his Modern self early on. The game does a fair job in illustrating that rather than having grown superior or inferior over time, Sonic and Eggman have moreso simply changed. When does this become awesome? When Eggman taunts the Sonics through the door leading to the Death Egg in the post, Modern Sonic decides to run his mouth as he is want to do. Classic Sonic just nuts up and goes for it. For more crowning, whereas Classic Sonic makes it through before the doors close, Modern Sonic just crashes into them like a bug on a windshield. In other words, he was faster.
The return of the G.U.N. military truck. This time it is equipped with giant saws and rocket boosters.
The Reveal that although up to this point in nearly every 3D Sonic game, Eggman got his ass kicked by the game's Eldritch Abomination, who went crazy and/or assumed itself as the Bigger Bad, not only was he able to somewhat control the Time Eater just by himself but, with the help of his past self, he turned it into a pilotable mecha! Talk about Took a Level in Badass.
The very fact Eggman while using Time Eater starts the game turning nearly the entire hero cast into statues. Just one cutscene into the game and he has neutralised almost all his foes except Sonic.
Defeating the Time Eater and both Eggmen, proves both Sonics to be badass.
Shadow's rival boss fight, in which you race for possession of power cores floating around the ARK. If Shadow gets two cores, he attacks you with multiple Chaos Spears and a blast of Chaos Control, accompanied by All Hail Shadow. If Sonic gets two, you need to smash meteors into Shadow whilst Live and Learn plays. The game appears to make a point of showing that Shadow matches Sonic point-for-point, too; he runs at almost exactly the same speed, and both characters will deflect away from each other if they get too close in Boost mode.
Remember the fight with Silver in Sonic the Hedgehog (2006) where it was freaking awful in every way? Well, you are going to love this one; No sarcasm! Silver does a much better job of putting across how powerful he is than any other game so far has, using his psychokinesis to its full extent. He throws multiple vehicles and various hunks of debris at you for most of the duration of the fight, fires the occasional psychic energy knife, and then tops it all off by chasing you down with a massive ball of twisted metal and debris, which Sonic can just barely keep ahead of. And he still throws other stuff at you during this!
The new version of the Perfect Chaos fight is this in general, as well as specifically for Sonic himself. He no longer needs to be Super to defeat the creature, and rather than fly across the flood-water, he simply hydroplanes across it by running at supersonic speeds. There's also that pretty sweet mix of Open Your Heart that plays for most of the battle.
Chaos, rather than just blasting raw power, is more cerebral this time round, blocking Sonic's way with loads of maze like pathways and obstacles, proving despite his role as an angry rampaging beast, he can test Sonic's brain as well.
Seeing both Sonics go Super Sonic at the same time, set to the insanely upbeat music from Sonic 4, like a happier DBZ.
The Badass Boast Sonic gives is a nice touch on top of that. "Time to scramble some Eggmen, SUPER SONIC STYLE!"
Classic Sonic punting Metal Sonic into the sky at the end of their duel.
Rooftop Run Act 2. Off the heels of a really hard level, check. Glorious music, check. Fast pace with enough challenges to remain fun, check. Rail grind that gives a view from hundreds of feet in air and then has you dive back into the town... check.
Crisis City. Given how the game it came from gets a ton of crap due to how rushed and buggy it was, it's awesome to see something from that game redeemed by having a fully developed and awesome game take a crack at it. They also made the music in both levels even better than in the 2006 game, and the music was one of it's redeeming features as it is.
They even manage to troll you and make it enjoyable in Classic.
Modern's level is also a great example of a level that is challenging, but not in a cheap way.
And an oft-overlooked one from the 3DS version. Right after both Sonic's and Tails' find the Time Eater, what to the Tailses do? They attack him head on, no questions asked! Sure, they failed, but their efforts caused the Chaos Emeralds to radiate with energy so the Sonics could transform into Super Sonic!
And they fare better than the two Sonics did before going Super anyway!
Modern Sonic is more of a Bad Ass than ever, as the ending cutscene/clip to the Egg Dragoon boss fight will show you.
Sonic running down a REALLY tall building in Speed Highway, complete with a herd of Speeders chasing after him, and accompanied with the "Goin' Down" music playing.
Some of the shortcuts in that level also require jumping onto a random helicopter flying above you, and getting back onto a rocket that Sonic just jumped off, making this one for the players too if they can succeed.
Eggmen smashing both Sonics prior to the final fight.
Your favourite Sonic music piece not in the Custom Music list in the game? For the PS3 version at least you can use the in-game XMB to play music from there, a feature that next to no PS3 game actually supports.
Unlocking Super Sonic as a skill! While it was first reintroduced for Sonic Colors, this game pushes Modern Sonic's use of the skill Up to Eleven! Now, in addition to normal bonuses Sonic's able to use a flight power similar to Chaos Control to tear through levels carefree. Watching him blast through what were previously incredibly difficult areas is absolutely awesome!
Classic and Modern Sonic's ranking animations while super are unbelievably badass, with nearly all of them being nods to poses and motions they've used in previous games.