WW2 and Vietnam for very obvious reasons are well known and covered in popular culture. There are decades worth of notable movies, TV series, video games, what have you about these two wars.
Yet with the notable exception of MASH, the Korean War has been almost totally ignored by US/Western popular culture as a subject for Hollywood war movies, video games and TV series.
Why is this so? The Korean War was fairly short; 3 years long, 1 year shorter than US involvement in WW2, yet it was very brutal and had a high casualty rate, in actual fact it had an even higher casualty rate (per capita) than Vietnam for American soldiers.
Here's the stats for US casualties in the two wars:
Vietnam, over the course of 10+ years (1965-1975) Total deaths - 58,209 / Combat deaths - 47,424 / Wounded in Action - 153,303
Korea, over the course of only 3 years (1950-1953) Total deaths - 36,516 / Combat deaths - 33,686 / Wounded in Action - 92,134
That's not even counting the millions of civilian deaths and the thousands of South Korean casualties and that of other UN forces such as the British.
A lot of Americans are completely unaware, for example, that the US military took on the Chinese military in full scale conventional war and it nearly escalated into WW3.
Despite it being classed as a mere "UN police action" at the time, it was for sure a real war... there were many notable aerial/naval battles and ground battles, such as Pusan Perimeter, the amphibious assault at Inchon, the battle of Chosin Reservoir where the 1st Marine Division and other US/UN units fought their way out of an encirclement of 12 Chinese divisions in -25 degree temperatures.
The South Koreans have released a lot of material/movies about this war such as The Front Line and Tae Guk Gi, but as i said it's been largely ignored by Hollywood and Western media.
Wouldn't it be good if, let's say, Tom Hanks and Steven Spielberg teamed up to make a movie or HBO miniseries on Korea, a'la Saving Private Ryan or Band of Brothers/The Pacific?
Certainly it must be more interesting than making the Nth WW2 movie, anyway?
I can't speak for why there have been relatively few major films or anything of the like that focus on the Korean War as I'm sure studios have a variety of reasons for why they haven't made any (ranging from lack of quality screenplays to simple perceived disinterest in the conflict by the American public).
However, I can try to shed some light on the reasons why it is often overlooked by many Americans.
First off there was the matter of the timing. Even though it is arguably the first major instance of conflict in the Cold War, it was still only a few years removed from the end of World War II. The war with Japan and Germany had affected nearly all facets of life for people in the states, as the nation's young men were mobilized and the bulk of the nation's industrial base being devoted to fueling the American war machine. Now, contrast that with the Korean War, which despite it's large amount of casualties, still had far less of an impact on American culture. Compared to the only "real war" many Americans had known prior to it, the conflict in Korea just seemed....insignificant, which led it to be marginalized.
You also have to account for the fact that the primary mode of carrying on the legacy of a conflict is through its veterans. The problem you see here is that by and large, most of the notable veterans of Korea had either already served with distinction in World War II (e.g. Douglas MacArthur) or would go on to do so in Vietnam (e.g. Creighton Abrams). Even President Eisenhower who oversaw much of the conflict is more associated with his command in WWII. Thus, in the decades that followed, you see a lot of veterans identify more with their experiences in other conflicts instead. And for those who did exclusively serve in Korea, it was simply a problem of being overshadowed by their peers from "The Greatest Generation".
Addressing why the Korean War is not regarded as prominently as the Vietnam War when looking back at the Cold War is another interesting topic, with a variety of explanations.
I believe the most credible theory is simply the massive difference in how the two wars were viewed domestically. The Vietnam War is famously synonymous with counter-culture, the anti-war movement, and the growing cynicism of the American public when it came to their view of the US government, yet when it comes to Korea, the home front seemed oddly silent.
While of course there were exceptions, it would seem that based on the high number of men who volunteered to fight in Korea (as opposed to being drafted), patriotism and the desire to fight was still quite high in the United States. Those who had been too young to fight in the European or Pacific campaigns a few years earlier now eagerly sought to prove themselves given this chance. There were others who, while lacking their enthusiasm, still felt a duty to serve (or to return to service) in order to protect South Korea. America was still coming to terms with its new position of power in global affairs, and the Korean War was to many Americans a test of their commitment to use this power to protect those who could not protect themselves. And in defense of this view, South Korea was (at the time of US entry into the conflict) in a very different position than that of South Vietnam years later. Despite the brutal rule of Syngman Rhee, you don't see South Koreans welcoming the prospect of being ruled by their Northern neighbors as much as you did in Vietnam. The North Koreans had launched a clear, conventional invasion of a sovereign nation, and it's clear that they would have easily conquered the whole of the Korean peninsula were it not for foreign intervention.
The somber and serious tone that is often prescribed to the Korean War is, I think, quite fitting especially after three years of fighting. Especially after the entry of China into the war and the settling down into a war of attrition, the men certainly became rough and jaded, but you don't see nearly as many lose faith in their cause as would be the case in Vietnam. And when they returned home, there were no grand celebrations in time square, nor were they met with widespread resentment. Perhaps it was the culture of the era, (precisely the culture that the youth of the 60s and 70s would push back against) but in the wake of Korea, most Americans simply sought to move on from it. It wasn't a great victory, nor was it a moral defeat. And so with seemingly nothing having changed as a result of the war, it was more or less like it never happened.
So in summary: It's often "forgotten" because it was overshadowed by World War II and the Vietnam War in the popular narrative of the US in the 20th century. It contributed very little to the US national identity unlike World War II (Ascension to global prominence) or Vietnam (Anti-War movement), and had comparatively little impact on the US public while it was under way.
Sources and Further Reading:
The Forgotten War: America in Korea, 1950–1953 by Clay Bair
Shadow of the Greatest Generation: The Americans Who Fought the Korean War by Melinda Pash