In some places, continuity has been wrecked. In others, it's fine. It hit Horde harder than Alliance, imo. Actually, if you care, I wouldn't even play Horde until I had cleared through NR, because you want to understand who Garrosh is, imo.
For instance, I do NOT recommend leveling a Forsaken character until you have cleared Northrend. Just don't do it. Their lore was grounded in their feud with the Scourge. So leveling up warring with Humans and Worgen is just going to be confusing, particularly when the plot elements require an understanding of Northrend lore. Plus, you'd spend so much time as a Hordie in Northrend dealing with the Forsaken campaign, it just doesn't make sense to spoil all of that.
But Night Elves? Continuity is even better now, barring slight time travel issues. In Vanilla, Malfurion had just mysteriously disappeared and you were living on this new tree and dealing with the loss of your immortality. That was really the extent of the story. Now, all you need to know, is that in the time between TFT and now, Malfurion had been lost in the Emerald Nightmare (which happened in the books anyway) because of an old enemy (from the War of the Ancients), but was being kept in the dream by the betrayal of then-Archdruid whatshisname.
Now Malfurion is out and the NEs are very actively countering Deathwing, Naga, Corruption, and Orcs. So, really, just where we left off + dragons. Biggest difference is that losing Thrall as Warchief vastly sped up deforestation under Garrosh--half of their most verdant woods are now gone.
So, really, their continuity is fine, since their plot was only loosely based on the earlier games (they didn't have loose plot threads to play with, beyond Illidan/Maiev).
Let's see... Human campaigns were similar--no real cliff hangers. Jaina is no longer the head of the Alliance. Well, she might be, but Varian Wrynn's political clout is so much higher than it really drowns her out. She also calls him "her King" at some point, which makes absolutely no sense at all (Jaina was born in Lordaeron, not Stormwind, and then became and Archmage of Dalaran, a completely sovereign city, and then founded her own nation state). Humans are back to warring with the Orcs, because the peace from the final bonus campaign of TFT died before Vanilla. All you really need to know is that the war has really escalated with Varian and Garrosh as the faction heads. Though, while I don't like Varian at all, he's significantly less blameworthy here than Garrosh is.
I'm missing a faction. Ummm... Oh. Duh. Orcs.
Yeah, no, Orcs have changed a ridiculous amount. Which is why I'd recommend leveling Alliance through Northrend before you go Horde. Thrall is out--he resigned as Warchief and a new (as of BC) character is in. The Orcs have, since that change in command, become vastly more militaristic. Though it's not all since that--the increasingly large-scale war efforts due to Illidan and then the Scourge had an influence, too. And Cairne Bloodhoof is dead, for reasons best left undiscovered until after Northrend. And Grom's story... well... Outland picks that back up and then carries in through NR and then into the new areas.
You'll really want, imo, to see the state of Orcs in both Outland and Northrend before you go to the modern day Orcs. You obviously don't have to, but I think you'd enjoy it a lot more if you are playing to see the story. You can really just start playing the Alliance stuff without batting an eye, but the Horde campaign will be like reading a book from the middle before later going back to the start.
Anyways, you all are horrible, @#%^ed up people