I had already done the Main Story Quest from the 2.1-2.5 patches. FFXIV requires them to unlock pretty much everything in Heavensward, save for the new race. However, I believe they've added gear to the Main Story Line that prepares you for the expansion, so you can skip the gear grind of the 2.0 endgame.
I've mostly been going through the Main Story Quest. I've completed the first zone so far. So far, it's pretty much just FFXIV, just more. Same style of quests and small stories. I did the first dungeon and first new primal. Both were pretty nice and reasonably easy. Thankfully, I'm a tank, so queues are instant. I've heard that queues for the other roles are pretty long. On the other hand, mobs in the expac seem to have a lot of hitpoints, so questing does take longer as a tank.
Heavensward is also extending the Job class stories, and giving out new abilities. I'm running out of hotbar space, though, so I'm not sure how useful this is, exactly.
In addition to the new zones and new stories, there are three new classes: Machinist, Dark Knight, and Astrologian. I haven't tried any of them yet.
The game also got a DirectX 11 version. I'm using it and it seems good. Though honestly, I don't really remember the old graphics, so I cannot really compare. The water looks really pretty now. The DX11 performance is fine for me, and my machine is about 3 years old (GeForce GTX 660 Ti).
Heavensward also introduced flying in the new zones. However, their solution to flying is the same as WoW's compromise. Each zone contains Aetheric Currents which you have to find and attune to. As well, certain quests also reward Aetheric Currents, so you have to complete those. Essentially, you have to finish exploring and questing in a zone before you can fly in it. However, since FFXIV isn't a seamless world, you unlock flying in each zone separately.
If you liked FFXIV before, you'll enjoy Heavensward. It's basically more of the same. If you didn't like FFXIV, I don't think Heavensward will change your mind.