Taking out Tywin and occupying KL are two different things.
Taking out Tywin is more of an immediate military accomplishment – it would involve the elimination of the Lannister host in the Riverlands, and essentially removing the Lannisters as combatants in the War of Five Kings, especially since the death of the liege lord of the Westerlands would be a huge morale/political loss on top of the military defeat.
Taking King’s Landing is more of a political accomplishment – it wouldn’t end the Lannisters as military combatants, Tywin would still have his army in the field, but without the Iron Throne (and presumably a king to put on it), political legitimacy would become an issue for him.
As to what Robb would do with it? I think there would be quite a bit of debate about that, similar to when he was acclaimed King in the North. I imagine the Riverlanders, being more proximate (and thus more exposed) to King’s Landing, would be pushing for Robb to sit the Iron Throne by right of conquest, whereas the Northerners would see that as demonstrating too much of a “Southron” inclination to Robb’s reign and would be more likely to want the Iron Throne left vacant (or take it home, in a reverse Stone of Scone, maybe). And the more politically calculating would probably want Robb to use it as a bargaining chip with Stannis or Renly or whoever.