Once both the United States and the Soviet Union had atomic weapons, the Cold War became a tense standoff in which each side knew that provoking a war would mean mutually assured destruction. An arms race in nuclear weapons developed, to the point that both nations possessed enough nuclear weapons to destroy not only one another but the whole world in the process. This caused much fear and apprehension among everyday people, to be sure. But it also kept leaders from actions that would lead to a direct war between the superpowers. The Cuban Missile Crisis of 1962, for example, showed that backing away from war was wiser than creating a nuclear clash with one another.
The two countries tended to pursue their Cold War agendas in other ways rather than by direct conflict with each other. They supported opposing sides in smaller wars that occurred (such as the Six Day War between Israel and Arab States in 1967). They sought to align other countries with them in their competing political and economic alliances.
Eventually, the US and the USSR began to talk with one another about reducing nuclear armaments because both sides recognized the enormous threat and danger posed by the massive arsenals they had developed.