I think it's important to remember that in Vietnam, the United States did very well militarily. However a large, vocal segment of the populace was very much against it. This coupled with the general ennui and terror at being thrust into a war that seemed to have no direct benefit to themselves, against their own will, completely destroyed all morale of the soldiers themselves. This caused politically infighting and instability, back home. In other words, the war wasn't so much lost as the leaders of a free democracy being unable to continue it, in a manner for their own and their people's best interest.
Korea was a slightly different matter in that the United States and S. Korea completely dominated the war early on (after recovering from the initial surprise and assault to Pusan). However, MacArthur's bloodlust and arrogance at being able to take China w/ Korea led to him not heeding China's warnings and rushing far too close to the Yalu River. Doing so along with statements regarding utilization of nuclear weapons brought China, with it's massive army of freshly prepared Revolutionary soldiers and Soviet air support, to full brunt against another group of soldiers who just wanted to go home. The Chinese wanted to get into the war, but couldn't without Soviet resources and air support; MacArthur gave Stalin an excuse to help and China an excuse to intervene.
Both times, the US lost pulled back because it was fighting on someone else's land and didn't have the people or soldiers behind the war. An apathetic offensive army away from home and a dissatisfied home front/civilian population are the worst ingredients for successful warfare. The defendants in both cases were fighting on their own land for something they very much believed in, which is a recipe for longstanding guerrilla warfare. Something notoriously difficult to win against.
EDIT: clarifying things
Even with the help of the China, how did Korea go from being Japan's whipping boy, to within 5 years, being able to hold the U.S. Military to a stalemate?