I would say there is a legacy to the Vietnam War in at least a few different ways, in terms of national sentiment, national politics, and military policy. That we spent so long fighting this war, lost so many people, and did not "win" the war have created consequences that have lasted this long and are likely to last longer.
The first legacy is one of protest, since the Vietnam War was one of the original issues that drew national opposition and protest, particularly among our youth. This war, as well as some other national issues, set the stage for a culture of protest, which endures to this day, in, for example, the "Occupy" movement or "Black Lives Matter."
Second, the Vietnam War led to the elimination of the draft. Yes, people still must register for the draft at the age of 18, but we now have an all-volunteer army. What happened in the Vietnam War was that the poorest of young men and a disproportionate number of minorities were those who had the highest risk of being drafted. Middle and upper-class young men were in college or had parents who were able to manipulate them either out of the draft entirely or into military positions that were relatively safe. Thus we sent our poorest and most vulnerable to die in the jungles of Asia, for what was clearly a useless and foolish endeavor.
Third, the Vietnam War led to the 26th Amendment of the United States Constitution, the amendment that gives 18-year-olds the right to vote. The nation came to see that sending our youth to die when they could not even vote on the matter was not acceptable in a democracy. Far worse than taxation without representation, this was death without representation.
Fourth, we can do a great deal of "what if" speculation on the course of history, without coming to any clear conclusions, but there is no question that this war nearly destroyed the fine legacy of Lyndon Baines Johnson domestically and led to a changing of the guard from Democratic to Republican. Nixon's administration was plagued by the war as well. We cannot say how anything would have played out politically absent the war, but clearly, it cast a long shadow on at least two administrations and consequentially, all subsequent elections.
Fifth, when those in power must assess foreign policy and defense, the Vietnam War often stands as a cautionary tale. When the United States began its wars in Afghanistan and Iraq, there were many who warned that either or both of these engagements could be Vietnam wars, sending our youth to die for a pointless mission, embroiling our country in unwinnable battles for years and years, in countries with cultures that we were abysmally ignorant of. It is my own opinion that such was the case, but you will find alternate opinions on that issue. The point is, though, that the Vietnam war is still used as a kind of touchstone or analogy in foreign and defense policy.
I am sure there are other legacies of the Vietnam war as well. As you study this sad period in American history, you may be able to see other ways it has an impact to this day.