The story did change drastically from the first game. Vanilla Destiny's story was a bust and was super hard to follow, The Dark Below was small and short with little explanation, House of Wolves was an improvement but still very little backstory. The Taken King improved greatly, but it was still missing some really important things, mostly characterization. Up until Rise of Iron we'd only seen Cayde-6 and Eris play a key and important role in the story. Rise of Iron did well, but failed on one important part. It explained a part of the lore very well through cutscenes, but I feel the story missions(apart from the last one) were done supremely poorly.
Destiny 2's story was very, very well done. From losing the Last City to getting your light back to gathering the Vanguard to take the fight to the Red Legion and Ghaul-- it was excellent.
Other than that, I have a real issue with some things that changed from the first game.
1) Shaders: The new shader system sucks. It was really nice being able to swap between shaders freely in D1 with no drawbacks, but the new system seems money grabby to me, and I don't like it.
2) Infusion: The new Infusion system is bad. It would be nice to infuse a power weapon with a power weapon or a helmet with a helmet but now you have to infuse it with the specific type. No more putting a 290 hand cannon into a 260 auto rifle, or no more infusing a duplicate Warlock helmet into a Hunter helmet you like.
3) Exotics: Don't get me wrong there are exotics that definitely feel exotic and look exotic, but most just feel like a slightly better legendary or look like a legendary. There's so few good exotics in D2.
4) Subclass Paths: I hate the restricted choices with subclasses now. 'Paths' are a great idea don't get me wrong, but changing from the freedom we had in D1 to a more restricted set of perks feels a lot worse.(The Nightstalker's paths are just awfully balanced)
5) Crucible: Crucible sucks still, and worse part is you can't pick the activity you want to play anymore. You're just randomly thrown into Supremacy, Control, and Clash. For me, I hate Supremacy, and I really only like Clash since I'm used to standard PvP gamemodes like that, so getting tossed into a Supremacy match when I want to play Clash is really annoying.
Other than that I really love the game. The Raid was done excellently and feels refreshing, the story was wonderful, Public Events are actually worth your time now, Worldly vendors was an outstanding idea, set perks for gear and perks no longer needing unlocked are amazing, characters getting the characterization they deserve, and more. I can't wait to see what the first DLC is packing!