Is everybody kind of overlooking something. While it might not actually destroy the planet, an all out global nuclear war would make the planet uninhabitable. The planet would survive, but we would be toast(ed).
I was about to make the same point with regards to the folks who pointed out that the Earth has survived collisions with Mars-sized asteroids, having a molten surface, etc... Yeah, the Earth survived in the *long* run, but it was uninhabitable for millions of years after each of those events, which needless to say would count as "the end of the world" for all practical purposes, from the perspective of anything living on the Earth, including us. Or to quote REM, "the end of the world as we know it", even if not the final end of this big ball of rock.
Cockroaches would survive, who would then re-evolve into democrats.