I don't know if you're going to get an answer that really fits the criteria for a good answer for /r/askhistorians (and mods can certainly delete this if it isn't good enough) but there are probably quite a number of factors that contributed to the differing public views between Korea and Vietnam.
The Korean war only lasted a bit over three years, while Vietnam dragged out for almost two decades. Keeping up public support for lengthy wars is not easy. The anti-Vietnam movement didn't really have huge numbers of supporters until the mid to late 60's, while the US had already been involved in Vietnam since 1955 (longer if you count financial assistance to the French).
The anti-war movement in the 1950's was weak due to strong anti-communist sentiment and fear of totalitarianism. The House Un-American Activities Committee was active and strong. Even socialist leaning liberal groups like the Union for Democratic Action rejected pacifism, which is perhaps understandable given that WWII could be seen a direct threat to western democratic ideals.
Also, the amount of media around the Vietnam War was far greater than during the Korean War. Average citizens would have a much better idea of the horrors of war from television than reading about it in a newspaper. Consider that in 1950 only 9% of homes had a television and was still less than 50% in '53, by 1960 it was over 87% and by 1970 95%. If you're looking at primary sources today, there is much more out there for Vietnam than Korea not just because of the differences in length of combat operations.
There were cultural changes too, of course. The counterculture and drug culture were much more popular, and open to questioning all aspects of society, which is not something you would have seen in the '50s when WWII was still a recent memory. Consider that basically a generation of kids had grown up without direct experience of the effects of war, and only the spectre of Mutually Assured Destruction that came as a product of the Cold War.
references: Television ownership statistics Info on the challenges faced by the anti-war movement in the 1950s.