How has the role of the United States changed since the collapse of the Soviet Union?
There have been two major changes in the role of the US.
First, the US is now the dominant country in the world. It no longer has to worry about a Cold War with the Soviet Union. Now, it is the only superpower and has more of a role to play in every part of the world (where once the Soviets "took care" of some areas).
Second, the US has to worry about more kinds of threats. Instead of just thinking about the Soviets, it now has to worry about "little" threats from more places. It has to worry about Al Qaeda and it has to worry about pirates in Somalia. It has to worry about conflict between China and its neighbors. Worrying about many small issues is a different challenge than worrying about one major issue.