I would say that its more about the cultural change than the war itself.

WWII is covered for obvious reason, and Korea came afterward. Korea in a lot of ways was the first fringe war of the Cold War, Vietnam was also part of it. Basically they all fell into the containment strategy.

The thing about the Korea was is there was very few things that were new or revolutionary about it. Jets came into widespread use, but over all it was fought, a treaty was made and its stayed.

Vietnam on the other hand represented a very turbulent time in American culture. Along with the assassination of Kennedy, MLK, RFK, the rise of the counter culture and all the other things people talk about in the 60s Vietnam came to represent a turning point in how America viewed our policies.

The 1950s were enjoying a post war boom period but the 60s were much different. Furthermore it is often considered a failure unlike Korea. Weather it was or had to be a failure or not can be debated, but in the public eye it was which represented the first real lose that the US ever had.

Basically, its not the war itself that causes it to be the focus, its the culture and its impact on American life that makes it so important to focus on.

To give a comparison, 50 years from now history books will focus a lot less on the first gulf war, than on the second because the first gulf war was very cut and dry, whereas the second was largely fueled by 9/11 controversial in its reasoning and tactics, and arguably caused a huge economic downturn that changed american society today.

The central reason the Korean War is skimmed over is because most history classes are poorly structured. I agree with you that it is skimmed over, but one absolutely needs to understand the Korean War in the Cold War context in order to understand everything that came after it in the proper context.

No Korean War? Probably a very diminished impact of McCarthyism, Nixon might not have become a Senator (and thus almost certainly wouldn't have become Vice President etc.) No Korean War? NATO quite possibly doesn't become as integral in binding the U.S. to Europe, the paranoia in the U.S. over aggressive global communism would have a chance to dissipate etc. No Korean War? Greater likelihood that Chinese - U.S. relations are less strained, and Chiang Kai-Shek likely loses influence at a heightened rate.

At least for now, Truman and Eisenhower (and their policies) are treated as much more boring and dull, despite the fact that those two (and Kennan, Nitze, and Dulles) effectively set the Cold War up on the American side. In order to understand Vietnam, you have to understand Truman and Eisenhower's policies. The Korean War is actually a wonderful case study, as both men were intimately involved with the conflict, and it shaped their understanding of communism, the Soviet Union, and America's role in the post-war world.

In order to get an idea of the Cold War, and essentially everything that happens on the international stage (from America's perspective), a detailed understanding of the context of the Korean War is absolutely necessary.