Azeroth is changed completely. Some zones are completely destroyed, some are shattered, some are flooded (Thousand Needles), some are ripped in half. Coastlines the world over are changed. Durotar is wrecked, along with Orgrimmar, which is torn in two. A new orc city is rebuilt over the course of the expansion. Gnomes retake Gnomeregan in an instance.
Malfurion comes back and helps activate the new Guardian of Tirisfal - Thrall. Thrall hands over leadership of Horde to Garrosh Hellscream while Thrall serves as a coordinator for both Horde and Alliance forces and players to combat the Naga and Black Dragonflight. Garrosh declares open war on the Alliance, and you can expect more Wintergrasp like areas on the old map. Due to some political manipulations, Cairne Bloodhoof gets painted as a betrayer and is murdered by Garrosh. Cairne's son takes over as Tauren chief.
For Worgen, the Cataclysm brings the Greymane wall down, revealing their secret -- that the worgen curse has ravaged the kingdom. They've managed to find a cure that keeps their human mind permanently but also keeps them in their worgen form. Seeing that they can no longer survive alone, the worgen residents of Gilneas rejoin the Alliance.
As for Goblins, Deathwing calls on the goblins as slaves like he has in the past. Their leader decides to go with Deathwing and enslaves his people because there's no crossing Deathwing and surviving. Fortunately, a large group manages to escape and join the Horde, knowing that they can't go it alone. See the pattern here?
The level cap is 85. New landmasses include islands that are raised as a result of the cataclysm, underwater zones, and the newly-redone old-world zones. Flying mounts will be useable in Azeroth. Many old-world dungeons are changed completely.
Half the point of Cataclysm is to remake the old quest lines, eliminate redundancy and give an overall narrative to all players, whether level 1 or 80. At any particular vanilla area, you can expect to see reconstruction/relief efforts, or all out war between Horde and Alliance NPC's. Wailing Caverns is going to be completely redone into a lush green tropical area, as a result of the druids working their magic to prevent it from falling into lava.
Leveling from 80-85 will be a MUCH bigger deal. Initially in 2008, they considered not raising the level cap at all, as making those quests takes a lot of manpower away from raiding content. But they are trying to make the story so grand and sweeping that ding'ing will actually MEAN something and not just be something you do.
Expect a couple of underwater cities to be on the backs of giant sea turtles and whales.
Make no mistake, the Old God content is not done. Since Netharion and Azshara both have ties to Old Gods, it is now clear that the SERIES wide antagonists of WoW are the Old Gods.
