Both are responsible for each of the many "proxy wars" fought between the U.S. and the Soviet Union. Vietnam was a proxy war, sure the North Vietnamese fought the South Vietnamese but who was actually fighting? The U.S. and Russia of course. In North Korea, the U.S. and the former Soviet Union fought. We fight against Communism and its ultimate goal of defeating Democracy. The Cold War was fought between the 2 global superpowers, the U.S. and the Soviet Union. The Soviets fought other countries. An example would be Germany (Oct. 2015.)
There were a couple of reasons for this. First, the Americans were an invading force, and the Vietnamese were fighting on their own soil. Second, the Americans were not willing to make an all-out commitment to win.