Someextra points to add to the many above. The Vietnam War was the first truly televised war for Americans. Instead of gathering by the radio and listening to motivating speeches and following the Allied forces' progress (as Americans did during World War II), Americans watched--almost in real time--what was going on in Vietnam. The images of bleeding, burned, mutilated, or dead soldiers were powerful and caused many to question if these losses (which now needed no imagination to consider) were worth whatever the purpose of the war was supposed to be.
Similarly, while the war was still in progress, veterans began speaking out against it publicly. Some told of the constant changes in the war's purpose (at least how it was presented to them) which caused them to question why they were in Vietnam. Others discussed the rules of engagement which differed greatly from other American wars. Instead of focusing on offense and victory, soldiers were told to not fire unless fired upon.
Finally, we have to remember that many of the men sent to Vietnam were baby boomers (the children of World War II vets), and many of their parents raised them in the idyllic setting of the 1950s when Americans wanted to put war behind them and give their children everything that they didn't have. This type of upbringing resulted in a much different type of American young person--not one who saw patriotism as all important or who was willing to go wherever his country called--but one who had already established different goals for himself or herself which didn't involve military service. This is not true, of course, of everyone of that generation (my dad voluntarily signed up and served a tour in Vietnam), but it does reflect the contrast between generations.