They competed in various ways, but primarily militarily and through space. Each built its military up to best one another, especially through the development of nuclear weapons. Space aspects, each side was in a space race to develop better and better space technology, which also partially related to military missile development.
Psychologicaly, it was terrifying for people. You didn’t know if a war would start the next day. Nuclear drills were common, which didn’t help either.
he space race was a competition between two cold war rivals i.e. the soviet union and the united states for supremacy in space flight capability. There were major psychological effects including anxiety, fear of devastating war, etc.