The Vietnam War changed the United States in many ways. I will look at three of the most important ways here.
First, it made the American people less eager to commit troops to combat overseas. The debacle in Vietnam left Americans worried (at least until 9/11) about the potential dangers of getting involved in protracted conflicts like Vietnam.
Second, it helped make Americans very suspicious of their government. The government tended to be less than completely open with the people about the war and this, along with Watergate, helped make Americans lose trust in government.
Finally, the war helped lead to a deep and abiding split among Americans in terms of politics and culture. It, along with the protest and the counterculture, helped to cause Americans to really become differentiated along cultural lines. On the one side were the relatively liberal people who worried about the use of military force and wanted to go against traditional societal values. On the other were those who maintained that the US needed to be strong and willing to use its power and that the old values were the best. This split continues to be seen to some extent today.