Expanded from an earlier answer of mine

The question of why Korea was divided and Japan was not is a tricky one with a lot to untangle here.

Firstly, the seeming permanence of Korea's division and the long-term division of Germany suggests that the wartime Allied powers planned for division based on geopolitical spheres of power. But the reality was these divisions were planned to be temporary, at least when the Allies were coming up with a schematic plan for the occupation of the Axis powers in the last years of the war. With regards to Germany and Austria, there was a wartime consensus among the Allied powers that both of these countries would need to be occupied for an undefined interregnum before a final peace treaty would unify them. The administrative divisions of these countries was less about carving out spheres of influence and much more concerned with the practicalities of administrative authority. The Soviet zone, for example, copied internal structure that the US Control Commission had come up with in 1944 because the Soviet government had not bothered to think that far ahead. Much of the division of Germany and Austria was predicated on the idea that this was an expedient and temporary measure.

The Cold War and growing distrust between the superpowers meant that temporary became permanent for the foreseeable future. While the responsibility for starting the process of turning the two zones into separate countries is a highly contentious issue in the historiography of Germany, one of the consistent features of early Cold War rhetoric was that both sides blamed the other for division and that German division was not their desired outcome. The idea that the two Germanies were in fact two separate countries took until the 1960s to become a mainstream opinion in public political rhetoric, even then there were still holdouts. While there was a diplomatic breakthrough on the Austrian question in the mid-1950s which ended the division and occupation of Austria, such a demarche was not possible with Germany. Not only were wartime memories of Hitler too strong to give sanction to unification, but both the GDR and FRG would likely sabotage any breakthrough that would leave one the loser in reunification.

Korea was a different matter than Germany. While the German question was one that preoccupied large number of intellectuals and policymakers, what to do about former Axis colonies was something that commanded far less attention in policy circles. While places like Formosa and Ethiopia were easily resolved by returning them to China or returning sovereignty to Jalie Selassie's government-in-exile, colonies like Korea, Somalia, or Libya were more difficult. While the Allies agreed that these areas should not be returned to the defeated Axis powers, there was no pre-existing political entity for the military governments to hand off sovereignty. These areas had been colonies for nearly fifty years and there were few indigenous political elites the Allies knew of that could readily take up ruling an independent state.

One solution the Allies employed was one used by the League of Nations after the First World War in which military governments would act as trustees for the freed nations while they developed their own political infrastructure. FDR had proposed such a UN trustee status for the whole of Korea, but this did not come to pass. The result was that the Allied occupation of Korea proceeded along a very haphazard manner. The December 1945 Moscow Conference issued a declaration that the US-Soviet military governments would encourage the development of peaceful, democratic parties in a unified Korea, but provided precious little in the way of actual structures to accomplish this. Dean Rusk would recall in his memoirs that the State Department decided on the 38th parallel as a division between American and Soviet occupation based on a map from an issue of National Geographic since Washington did not have any decent maps of Korea. The 38th parallel became a consensus choice for both superpowers as it neatly divided the country while ensuring the US zone received Seoul.

Yet the occupations of Korea soon went on separate trajectories with a good deal of American disinterest in the details of occupation and the joint administration of the peninsula fell apart. While the Soviets showed an interest in sponsoring a Korean Communist Party in their zone, American support for Rhee and other Korean politicians ranged from indifferent to soft support. Yet, despite the development of two separate political zones, the raison d'être for parties in both political zones was to lead a unified Korea. Like works on German disunity, the historiography on responsibility for the Korean War is quite complex and multifaceted, yet there was an overall desire on both superpowers' part to have a united peninsula, albeit on strategic terms favorable to them.

In more than a few ways, the course of the Korean and German divisions are reversed images of each other. While there were multiple solutions to the German Question bandied about in 1945, Korean policy was often shaped in a vacuum. While the charter generation of postwar Korean politicians stressed unity and worked towards that end, to disastrous results, a good many German politicians on both sides of the Iron Curtain recognized the geopolitical realities of the time and accepted division. By the same token, both superpowers nervously eyed each other's moves in German affairs, there was remarkably little in terms of actual Korean policy for the superpowers to analyze. One of the reasons why Stalin backed Kim Il-Sung's invasion was that the US had omitted Korea from a policy paper on its defense zones in East Asia. Finally, one of the salient features of German division was that while the Soviets had a roughly pliant satellite Eastern Europe and German communist party, the Soviets faced in the form of Mao an upstart communist leader that the Koreans could use as leverage against the Soviets for their own purposes. Soviet policy towards Korea not only had to contend with American reactions, but also their fluctuating relationship vis-a-vis the Chinese.

Japan was a somewhat different matter entirely.

Although the US presence looms large in Supreme Commander for the Allied Powers (SCAP)'s occupation of Japan, the occupation of the home islands was not a US-only affair. There was a British Commonwealth Occupation Force (BCOF), dominated largely by Australians that operated under GHQ's authority. The BCOF included contingents drawn from the British Empire and the Commonwealth with headquarters in Kure. But the BCOF was largely a failure; it was an expensive undertaking without any true mission for it. Still, the BCOF was not an analogue to the division of Germany. This was by design as the US took active measures to ensure that US authority would prevail over the home islands.

The decision to transform the occupation of Japan into an American-run show was not exactly straightforward. The Pentagon initially envisioned a division of the Japanese archipelago into multiple occupation zones. The Joint War Plans Committee produced one plan, JWPC-385/1, entitled "Ultimate Occupation of Japan and Japanese Territories," on 16 August 1945. JWPC-385/1 envisioned splitting the home islands between the US, the Soviets, UK/Commonwealth, and China with Tokyo split between the four powers. The USSR would receive northern Honshu and Hokkaido, while the Americans governed Honshu, the RoC and US held the Kansai region with Chinese control over Shikoku, and the UK governing southern Honshu and Kyushu. As with the occupation of Germany, this occupation mixed geographic logic- the Soviets in the north, for example- with a somewhat haphazard and arbitrary division lines. JWPC-385/1 planned for a staged occupation with a degree of collective leadership that would terminate within a few years with a final peace treaty between a reformed Japanese government and the Allied powers.

JWPC-385/1 though was a dead letter as the Pentagon drew it up. Both the State Department and President Truman did not want to replicate Germany in the Far East. Truman would later recall in his memoirs:

I was determined that the Japanese occupation should not follow in the footsteps of the German experience. I did not want divided control or separate zones. I did not want to give the Russians any opportunity to behave as they had in Germany and Austria.

Although Truman's claim has to be seen within the context of the early Cold War and an ex post facto justification of his actions, he was moving by August to ensure the Americans would run the occupation of the home islands. He already refused a Soviet request at Potsdam for a Soviet postwar occupation of Hokkaido. The State Department via the State-War-Navy Coordinating Committee (SWNCC) in the meantime had drafted SWNCC 70/5, viewable here, on 13 August which outlined much of what emerged postwar as SCAP. This document was the culmination of thinking within the State Department which stretched back to 1944 in which the occupation of Japan needed an American control that was unquestioned. SWNCC had a leg up on JWPC-385/1 in that MacArthur's staff was already opposed to a zonal division and had a preexisting contingency plan for occupation of the home islands- Operation Blacklist- in case Japan suddenly collapsed. So the State Department have a powerful ally within the Pacific to argue against the Pentagon's JCS factions favoring a zonal division. Truman ultimately sided with the State Department and SWNCC 70/5. Truman had made his decision known by late August 1945 and endorsed the SWNCC plan for Japan.