One important aspect is that the Korean War was comparatively short, lasting only 3 years, whereas US involvement in the Vietnam War lasted over a decade. The period of time between the Tet Offensive alone and the pullout of US troops was over 5 years.
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Why was the Korean War such a success, and the Vietnam War such a failure?
In addition to the timeframe and scale aspects mentioned by other posters, I would like to point out that the involvement of the media was radically different. Korea was essentially only covered by print media, and at that, there was a very cozy relationship between the media and the US government. It has also been mentioned by other posters, but 1950 was the real start of McCarthyism and any anti-war sentiment could easily be perceived as sympathy for the communists, which could easily leave you ostracized from society.
In Vietnam, on the other hand, there was not the same level of political/social control at work. At least as importantly, field journalism and imbedded tv news crews had become a reality, so Americans could actually see the terrible things that were happening, which does a lot to change the way that they felt about the war.
The Korean War took place in the height of the Red Scare in the west and at one of the highest points of activity against communist forces, by the time the Vietnam War escalated this support had declined significantly.
The forces in the Korean War technically were representatives of the UN approval of international forces being used to defend the republic in the South. The Vietnam War never had this level of approval and was a unable to gain international support in large scale for the South Vietnamese.
The increase in media coverage during the Vietnam War especially TV and color photography brought the war to a new level for the general population who had previously been mostly removed from wars in the past by state censorship or lack of availability.
In Korea the US was acting upon UN mandate ( United Nations Security Council Resolution 84 ); in Vietnam it was acting upon invitation of unstable governments (after the coup against Ngô Đình Diệm) without clear legitimacy, the politics of South Vietnam were a mess with officers in charge, all of them grabbed power by means of a coup, so people would ask: who is the good guy here?
The main line of reasoning for the Vietnam war was to prevent the Domino theory - a fall of Vietnam to communists would cause the collapse of other Asian countries, etc. ; the long line of reasoning of this theory was not the same as the danger that comes from having to face an invading army (like in Korea).