The easy-but-wrong answer is "deterrence." In the case of proxy wars, it is not clear that deterrence actually explains why the US (or USSR) didn't use nuclear weapons in a limited way. A related answer, "escalation," does work in Vietnam to an extent but not Korea (the US policymakers did fear that if they used even tactical nuclear weapons in Vietnam, it would encourage greater participation from other powers and possibly encourage the use of tactical nuclear weapons against American military targets).

But the broader question is a really difficult one. How, and why, did the "nuclear taboo" get established after World War II? The US was crucial to its creation, if mainly because there was a long period (1945-1960 or so) where the US could have used nuclear weapons without expecting significant retaliation in kind. (The Soviet Union had nuclear weapons starting in 1949, but their ability to use them against the US mainland was extremely limited until they had ICBMs to deploy. The US by contrast had by 1960 ringed the USSR with friendly airbases filled with nuclear-armed bombers.)

The interesting answer, in my mind, is that the President who perhaps did the most to establish the "taboo" is Truman, the same one was the one to initial use the weapons. He resisted normalizing the use of the atomic bomb very vigorously (even physically denying the military access to the weapons, despite their protestations), and he considered them to be something "special" well beyond what his military advisors thought at the time. Eisenhower started his term being much less worried about the use of the weapons, but ended it with a very different understanding. By the time of Kennedy the taboo was basically established as a strong norm and while the idea was floated occasionally regarding, say, Vietnam, it is not clear that it was taken very seriously.

Korea is a very interesting case. If Eisenhower had been President during the bulk of it, would nuclear weapons have been used? I think very possibly — and that would have been in accordance with most US military thinking at the time. Truman and the Joint Chiefs considered and rejected the possibility of use largely on the basis that it would alienate the international community and the United Nations; they argued that the political cost would be higher than the military value would be worth.

A similar logic ruled the day in Vietnam as well. World War II might give one the impression that nuclear weapons "end wars" but it is by no means clear that this would be the case, especially when fighting against a largely decentralized and highly-mobile enemy. A dawning realization that the US actually benefited the most from the nuclear taboo started to take hold: the US is actually quite vulnerable to nuclear weapons, both in the strategic (cities on coasts) and tactical (our deployments rely on massive bases or ships to act as command and supply centers, and these are easy targets for a tactical nuke) senses.

Anyway. There's more that can be said here — Nina Tannenwald's The Nuclear Taboo is an excellent study of the question.