The North would cause a lot of catastrophic damage to the border region and the Seoul metropolitan area, but they'd shoot their wad early in the game. The overwhelming U.S. and South Korean superiority in air and naval power as well as the much better quality of their ground equipment would result in an Allied victory...but it wouldn't come quickly or cheaply. A second Korean War would make Iraq or Afghanistan look like quiet peacekeeping assignments.
Yeah, basically, Seoul would be obliterated in a few hours, and then all of North Korea would be annihilated shortly after. It is not a war that NK can win in any way, shape, or form. While they can devastate the South in a first strike, that's really their only advantage.
Look at it as though it were a hostage situation: North Korea's got a gun to the South's head. If NK wants to live, the worst thing they could do is pull the trigger, because the US has a dozen snipers with a perfect kill shot ready to go and if SK is done, there's nothing left to bargain with NK over.