What is the difference between Proxy war and cold war?

The term cold war refers to the strained relationship between two countries or two power blocks where neither rival participates directly at war with the other where each of the rivals tries to soften its opponent. Historically the term refers to an adversarial relationship existing between the United States and the USSR during the period, beginning of the end of WWII and the fall of the Soviet Union.

During this time, no direct war between the two took place. A proxy war is a conflict between two powerful nations where neither party directly attacks or acknowledges military hostility toward the other.