One of the most cultural important influences of the Vietnam War was to inspire the spirit of rebellion and protest that became such an integral part of the 1960s. The growth of the counterculture movement and the spirit of free expression found a home in the opposition to the war. As the conflict escalated, individuals who were a part of the counterculture sought the forum to voice their dislike for the conflict. The social structure of the family also saw a level of change as the war became perceived as engineered by the old against the "flower of the youth," as most of the soldiers sent to fight and die were young. This helped expand the counterculture movement to be a generational one. In terms of lifestyle, the Vietnam war was seen as an example of conservative American values. The counterculture sought to define themselves in opposition. Short haircuts and "acceptable" dress was challenged with long hair and exploration of different fashion statements. Drug use became embraced as a part of the questioning and exploration that was antithetical to the conservative values of the time. Some of the most enduring music of the time period were either the protest songs of the counter culture or the music inspired by the exploration of this culture. Woodstock became the ultimate statement of counter culture, setting itself in stark opposition to what more "traditional" values would have espoused.
The Vietnam Conflict (was it ever a declared war?!?) may have been the catalyst that altered social structures, lifestyles, and created the counterculture, but the root causes for the massive changes are deeper. As earlier posts suggested, the 1960's hippie movement originated from the 1950's beatnik movement, and here we have to draw an historical line. The reason the 1960's were so turbulent was due to the turbulence of the 1940's -- namely, the Second World War.
For those who grew up before the war, certain social values and structures were in place, which by the 1960's were considered conservative. The 1960's showed the emergence of that first post-war generation -- those who knew nothing of how life had been before the war (since that's when most of them had been born) and questioned every aspect of the established culture. Furthermore, these first baby-boomers, all near the same age, made their points by their sheer numbers. This explains the more narrow beatnik movement compared to the widespread hippie movement. Had the baby boom not happened after World War II, the "flower power" generation would not have been as unified and as forceful. Additionally, had Vietnam not served as the focus for all they found wrong with the culture, the hippie movement and its impact would have been much less.
Since the U.S.'s involvement in Vietnam was a part of the Cold War struggle, (namely Capitalism vs. Communism), the effect it had was initially ideological. Just like the war today, the social effect was a polarization with the counterculture emerging primarily on the left and the "silent majority" on the right. Reasons that the counterculture seems more prominent during the Vietnam war as opposed to anti-war movements now: there was a draft, the war coincided with the Civil Rights movement, and Vietnam was the first war that was immensely televised, so this was the first time U.S. citizens really saw images of war as they were happening.
With the large death toll of U.S. soldiers as well as the much larger toll of Vietnamese (including massacres of innocent civilians), the outrage against the war intensified. The anti-war movement was a statement against what was seen as U.S. imperial/ideological expansion. The counterculture, already militantly engaged in civil rights and peace movements, found the anti-war movement to be philosophically, politically and ethically in sync with what they were already doing.
In addition to the very good points made above, I would like to add just one.
The Vietnam War helped to push the counterculture farther toward a hatred of authority. This had already been happening to some extent as a reaction to what the counterculture people felt was the overbearing authority of their parents. But it grew as a response to the war. The people of the counterculture felt that the war was an example of how blind obedience to authority was ruining our civilization.
It was in reaction to this perceived excess of authority that the hippies tried to form communes and other social structures that lacked any authority figures. (Of course, that didn't always work so well as some people fleeing authority ended up under the authority of people like Charles Manson...)
It is important to note that the 60s counterculture had its roots in the 1950s beatniks. By the 1960s, the counterculture became a mainstream movement. The Vietnam War provided a perfect forum for the movement to solidify behind a cause and it then moved onto college campuses and grew in epic proportions.
By its nature, the anti-war movement was a rejection of authority especially when one considers that the soldiers expected to fight were not volunteers, they were conscripted. The modern day wars differ from Vietnam in that core fact, the current army is an all-volunteer army and although some may argue against the war in principal a soldier who volunteers for military service is similar to a police officer in that there is an inherent risk with the job. Ultimately it was the draft during the Vietnam War era that changed social structure.
There are a number of factors you could look at relating to social strcture and lifestyle of the 1960s counter-culture. First, consider the impact of the war being broadcast on television. For the first time, Americans in their living rooms were exposed to the reality of war. Body counts were published. The impact of this was horror, shock, and disgust for many people which fueled the rebellion.
You can't ignore the music of the counter-culture. There were protest songs, such as Country Joe McDonald's "I-Feel-Like-I'm-Fixing-to-Die" or Crosby, Stills, and Nash who wrote several protest songs based on Nash's work with Vietnam Veterans.
You could also look at the effects of protests against the war and their outcome. Consider Kent State and the violence against students in 1970 or peaceful protests like the "Human Be-In" at the Golden Gate Park in San Francisco in 1967.
And of course, there is the rampant drug use both by soldiers and protesters, ranging from marijuana to psychotropic drugs. Wide-spread use to escape reality or to seek enlightenment through pharmaceuticals lead to tighter drug laws in the US in the 1970s and 1980s.
Hope this helps!