Given the effectiveness of nuclear weapons in quickly ending the war with Japan in 1945, why were nukes not used during the Korean War less than 10 years later?

It's a complex and interesting question. The Korean War was, in many ways, the first real test of the "nuclear taboo": the US had a relatively large atomic arsenal; the Soviet Union technically had nuclear weapons, but would have had a tiny arsenal and not the delivery mechanisms to go beyond Europe; the US felt that its postwar credibility was on the line with regards to Communist aggression; and so on.

And indeed, during the war many top US military and policymakers talked quite openly about the idea of using nuclear weapons in the conflict. General MacArthur in particular seems to have had no reservations about using the atomic bomb.

So why didn't they? There wasn't any big, singular reason. There was no "big showdown" over using the bomb. Rather, you had a lot of things that sort of resulted in a lack of use. These included:

Truman himself was very dubious of military use of the bombs. During this period, the military literally had either no or very limited access to the bombs themselves — they were kept in civilian custody (a small number were approved for transfer to Guam in 1951). The military repeatedly requested the bombs; Truman deliberately and repeatedly prevented them from having them. In various private conversations he made it quite clear he did not think they were "military weapons" that should be treated lightly. As he told the head of the Atomic Energy Commission in 1948:

I don’t think we ought to use this thing unless we absolutely have to. It is a terrible thing to order the use of something that, that is so terribly destructive, destructive beyond anything we have ever had. You have got to understand that this isn’t a military weapon. It is used to wipe out women and children and unarmed people, and not for military uses. So we have got to treat this thing differently from rifles and cannons and ordinary things like that.

The Joint Chiefs of Staff rejected the idea in late 1950 for the following reasons: 1) the US didn't have enough weapons to deter in both Asia and Europe at the same time (and considered Europe more core to US interests); 2) there were few useful targets in Korea for the atomic bomb; 3) and, importantly, they feared that the bombs would not be decisive, and as a result diminish their role as a deterrent. The latter is perhaps the most interesting: why wouldn't they have viewed the bombs as decisive? The US military, as reflected in things like the US Strategic Bombing Survey of 1946, really didn't think that the bombs ended the war with Japan — they thought, at best, they were the "icing on the cake," a cake that had already been built with massive bombing raids, blockade, and the threat of invasion (much less the Soviets entering the war). That is, they didn't really believe that you could just replicate the end of World War II with them. And if they didn't produce a decisive end, what then? This is a nice point — the "nuclear taboo" against using the bomb, in many ways, raises the importance of the nukes as a deterrent. If nukes are "just another weapon" then they cease to be as much of a trump card.
After the Chinese entered the war in force, there were calls to reconsider the nuclear option. But the State Department opposed it strongly — they saw it as a political problem, internationally. They argued that conventional and atomic weapon use would be viewed very differently even if they had the same outcome in terms of military or civilian deaths. Indiscriminate use of nuclear weapons would hurt US efforts with its allies, and undermine the US "moral position." Others in the military also worried that even legitimate "tactical" use of atomic weapons (say, against troop concentrations) might bring the USSR into the war.
This debate went back and forth over the course of the war. Because these issues were, to various degrees, being discussed in the press as well, it led to a large opposition from third world countries in the UN about US use of nuclear weapons in Asia (again). It is worth remembering that the Korean War was a UN peace action, so UN approval was not exactly unimportant. And the Cold War, though often portrayed as a conflict between superpowers, was also largely concerned with courting the third world as well.

In the end, how should we view this? I think it's worth noting that it really could have gone in either direction. What was the restraining force? Undoubtedly Truman in this case — by 1950 he was very averse to the use of nuclear weapons, and dubious of arguments about their military utility. One can argue about why that was (I suspect it was maybe not quite remorse, but a deep, unacknowledged discomfort with the way Hiroshima and Nagasaki went down), but Truman had a personal aversion against this. And without his approval, it wouldn't move forward — again, he had deliberately centralized the question of the use of atomic weapons, and most of the physical weapons themselves, in the office of the president, in a way that was not the case in World War II.

(It was only under Eisenhower that the military got access to the bombs themselves again, even though they were not to use them without presidential authorization.)

If Eisenhower had been president in 1950, would he have authorized their use? It's not impossible — he started his presidency much more bullish on the use of nuclear weapons for tactical purposes. It's of course impossible to know. But there's no overwhelming logic that kept the bomb from being used in Korea: there were strong arguments on both sides, but ultimately none strong enough to overcome Truman's aversion.

For the sort of canonical discussion of this question, see Nina Tannenwald's The Nuclear Taboo: The United States and the Non-Use of Nuclear Weapons Since 1945 (Cambridge University Press, 2007), chapter 4 ("The Korean War and the Emerging Taboo").