The major change in America's role is that it was no longer competing with the Soviet Union for world domination. Instead, it became the sole superpower in the world and came under greater pressure to act as "the world's policeman."
During the Cold War, the US was mainly trying to fight communism. Its role was to promote the interests of the free world and to serve as the leader of the free world against the Soviets. After communism, the US was the only superpower. Its role now seemed to be that of keeping peace in the world as a whole. This is why, for example, the US got involved (soon after the end of the Cold War) in conflicts in places like the Persian Gulf, the Balkans, and Somalia.