They were both extremely similar in premise; a battle of North vs South, Communist vs Democracy, both overseas not near the US. Korea veterans were not regarded very fondly, yet there's a lot more respect for those veterans from Vietnam. Why?
There have been 3 reason I've read.
The war was not an "American" war where we were fighting against a direct threat to us, but a "U.N. police action", which had a psychological impact on the public of making it something short of a war like WW2 was. Think of the U.S. intervention in Yugoslavia or Panama. It was a conflict you could actually tune out of if you wanted. There was no air-raid drills, rationing, scrap drives, swing-shifts at the Lockheed plant, and no Rosie the Riveter. Sure there was conscription, but not as many guys to make a noticeable impact at home, and a great deal of them were guys still in the military (like my grandfather) from the end of WW2
It didn't end in a clear-cut victory. Muddled outcome of the War of 1812 aside, Korea was the first conflict in American history that did not end in a clear "we won and they lost" conclusion. As "Patton" said at the beginning of the film, America loves a winner and cant tolerate a "loser". Since the war ended in a stalemate after we were humiliatingly pushed back from the Chinese border with victory in easy reach, it was a war that people were happy to take the armistice and move on, and as a result, didn't celebrate what was seen as a inconclusive outcome from guys that didn't "win".
WW2 fatigue. The nation had just had enough of giving so much attention to wars and depression, and wanted to move on to living the good live of the post-war economic (and baby) boom.
As a kid in the 80s, I had not even a vague idea of what "Korea" was. Saw tons about Vietnam and WW2 though. The vets from either were always the center of discussion because many of them served in Korea after WW2, or Vietnam after Korea, and simply saw those conflicts as having a greater impact on their lives. Or they were in either WW2 or Vietnam, and not Korea, and thus Korea was just some thing they didnt focus on too much because they were too young at the time, or were still shaking out their World War experiences.
That would explain why it took years to get a Korea memorial. It just keeps getting lost in the shadows of those bigger conflicts.
[Bear in mind, I know Korea was a UN operation with lots of Allies, this is just an American perspective] Without going too deep into the politics of any of the wars, there's quite a few reasons
The Korean War happened in 1950-1953, in between WWII and Vietnam, WWII obviously being the most important war in recent history, and Vietnam having the huge cultural impact on the US that it does. Vietnam was very a very decisive war, many Americans and the rest of the world were vehemently against the war.
Most education courses about the Cold War (in my experience) tend to focus on Vietnam more than Korea, the Korean War is given very little attention, despite it's importance during the early stages of the Cold War.
Hollywood/cinema/media attention. Like it or not, it has a huge influence on public knowledge and perception. There's been several decades worth of movies and TV shows based on or in WW2 and the Vietnam war, up to this day.
If someone says "WWII film" to you, you probably think of 'Saving Private Ryan', 'Fury', 'Inglorious Basterds', 'The Boy In Striped Pajamas', 'Schindler's List', 'The Thin Red Line', 'The Code Talkers', 'Flags of Our Fathers', 'Letters from Iwo Jima' Or HBO's two famous TV series; 'Band of Brothers' and 'The Pacific', and their new upcoming 'Masters of The Air' Maybe if you're a bit older you think of 'A Bridge Too Far' or John Wayne's 'The Sands of Iwo Jima' and 'The Longest Day'.
If someone says "Vietnam movie" you probably think of 'Full Metal Jacket', 'Platoon', 'Apocalypse Now', 'We Were Soldiers', etc. You get my drift. WWII and Vietnam have a lot of movies.
Now if someone says "Korean War movie", my guess is that you don't know any, or if you're older, have a vague memory of MASH which ended decades ago. Maybe if you're a particular fanatic, you might watch the low budget South Korean movies on the war, but the point is, Hollywood and American film-makers don't make movies and TV series about Korea.
Unlike WW2 and Vietnam, there was no decisive victory. The US pulled out of Vietnam, whereas the US and the Allies outrightly won in WW2. Korea was and is a ceasefire that exists to this day. Draws are less memorable than winning or losing wars.
Just my personal theory, but part of it, I think, is because Korea was fought mostly by the same generation that fought in WWII. A lot of WWII veterans went on to fight in Korea, such as Capt. William E. Barber (USMC Medal of Honor winner, fought in Iwo Jima and Chosin Reservoir). It only started 5 years after the end of WW2, so to the untrained eye especially, photos of soldiers in Korea can very easily be mistaken (and frequently are) for soldiers who fought in WWII as the equipment was mostly the same. US GIs and Marines in the infantry fought wearing the same uniforms they did in WW2, they carried the same M1 Garand and M1 Carbine rifles of WW2 infamy, etc. The only notable difference, I think, is in aviation. The Vietnam era GI is iconic in his own right, with his olive drab or tiger stripe uniform, and M16A1 rifle fighting Vietcong guerillas in the dense jungle or across paddy fields. Also a lot of big WW2 names, like General Douglas McArthur were involved in Korea (Korea actually ended his career). My point being, photography plays a huge part in our memory and consciousness.
In my opinion it's a great shame that the Korean War is largely ignored by historians and the media, I think a movie or HBO miniseries about the Korean War would be awesome. There is a nice memorial of the Korean War in Washington DC, but even so it's such a shame that the vets of the war are largely ignored.
There were many great, bloody battles in Korea, such as the Pusan Perimeter, Chosin Reservoir, Inchon etc that the US public just have little to no wareness. It's not like Iwo Jima or Normandy which pretty much every American has heard of.
Also what's bizarre, is that Korea was America's third most bloodiest war of the 20th century on a per capita basis after WW1 and WW2.
These are the casualties of the Korean War, in only 3 years - or 37 months exactly
Total US casualties: 128,650 Total deaths: 36,516 KIA: 33,686 WIA: 92,134
That's 45 deaths a day, or nearly 3,500 deaths a month.
The Vietnam War spanned from the early 60's, to when the first advisers went there, to 1975 when the last US troops high tailed it from Saigon.
For Vietnam
Total US casualties: 211,454 Total deaths: 58,209 KIA: 47,424 WIA: 153,303
That's 11 deaths a day, or nearly 1,200 a month.
As you can see, Vietnam was horrific, but Korea war much more intense. Korea was a conventional war against the state North Korean and Chinese militaries, on land, sea and air. Vietnam was largely a COIN war.