Why didn't they nuke each other?

Mutually assured destruction.
The threat of total annihilation was too great. Even if one of the superpowers struck first, there was still a very good chance that the other side would not be so totally incapacitated that they could not inflict a potentially devastating retaliation. While in our simulation, there was a lot of angry noise being made, I think that in the actual Cold War, high tensions did not necessarily mean an unwillingness to talk. The concept of MAD entails the exposure on both sides of their weaknesses to nuclear attack, and neither the USSR nor the USA could afford to be entirely gung-ho with their nukes so long as they were vulnerable. Bluffing and rabble-rousing talk might have led them to the brink of nuclear war, as in the case of the Cuban missile crisis, but in the end, the Russians and the Americans backed down. The stakes were too great, and the crazed ideology not strong enough, for either side to sacrifice civilians, cities, and the stability and well-being of their country for the destruction of the other.