Both sides had enough nuclear buildup that if one directly attacked the other they could destroy the world. So achievements like the space race and smaller wars fought in the middle east coupled with the fall of the Soviet Union determined the winner. It was basically an endurance test.
Nuclear war duh. That's why it was called the Cold War in the first place there was no fighting to begin with. It was just a series of threats and who could have the best things like the Space Race.
