USSR and USA both knew that if they would engage in a full scale nuclear war, it would simply lead to annihilation of human life.
Such an event was termed as “MAD” or “Mutually Assured Destruction”
Both USSR & USA didn’t want this catastrophe to happen. So they didn’t allow it to turn it into a ‘hot war.’
But both superpowers participated in proxy wars like in Vietnam, Korea & Afghanistan to prove their might. In these wars they were not engaged directly but were helping local rebels or the ruling body with weapons and technical assistance.
The Vietnam & the Korean wars were won by the USSR backed side. While USA backed mujahedin fighters managed to win over Soviet backed Afghan government in Afghanistan.