You really need to do the quests on both sides to get the full story.
This is true. However, the theme of this expansion is meant to bring tension between the factions back into the fore. Hence, Thrall is gone, replaced by the warmongering Garrosh, the worgen - arch enemies of the forsaken - have been introduced to the Alliance, the goblins - who had their ship destroyed by the Alliance - are now a part of the Horde, and so on.
There is also tension within the factions, as the dwarves don't have an entirely stable government yet (and Stormwind's SI:7 attempted unsuccessfully to assassinate her when she made her claim to the throne of Ironforge), the orcs (or, more specifically, Garrosh) have banished the non-orc races to the outskirts or slums of Orgrimmar, Sylvanas continues to make moves against Garrosh's orders (at least he's smart enough to be wary of a universal, life-destroying plague), and so on.
The idea right now is that the comfortable, relatively peaceful time people spent in Dalaran with members of both factions (or, to a lesser extent, Shattrath - where people could be of the same nominal allegiance as members of the opposite faction) are gone. The portals in Shattrath and Dalaran were removed partially because of this, so that neutral areas would no longer be the main hubs for all players of both factions. Even though Deathwing and the Twilight's Hammer are a common enemy (as Arthas was in Wrath, and the Burning Legion before that), Cataclysm is meant to reinforce the notion that the factions, despite being the "enemy of my enemy" to each other, are still at war.
I believe the expression goes: "Putting the 'War' back in 'Warcraft'."