Shortly after World War II, the French were trying to reestablish their rule in their colonies/mandates. From what I've read, the British lobbied the US to support the French in the French's desire to take back Indochina, in part because the British Foreign Office didn't want the US to undermine European colonial empires since they were afraid of blowback on their own empire. During the same period, though, I've read that they pressured the French to pull out of their mandate in Syria.
What was the difference between these two situations that made the British act in such different ways?
There were several major strategic and legal differences which made the two situations functionally distinct.
Firstly, Syria had been acquired by France as a League of Nations mandate from the Ottoman Empire. It was intended to be a stepping stone towards eventual independence, and recognised as such by all international signatories, including a legal requirement to never annex the territory. On the other hand, Vietnam was a colonial possession, and there had been no plans to relinquish a diminish French presence in the country before WWII.
Secondly, Syria had already been granted independence in 1936, which was later deferred as threat of war escalated. When the war concluded in 1945, France was legally obligated by international law to obey the terms of the treaty.
Thirdly, Syria and Lebanon were one of the founding members of the United Nations in October 1945. This was problematic because the UN Charter did not allow nations to be officially subordinate to each other. The Soviet republics circumvented this by being legally independent republics, but France could not feasibly claim that Syria and Lebanon were both UN members and also trustees of France. Vietnam was not a member until 1977.
Fourthly, the UK had no strategic imperative to promote French interests in the Middle East, when it was itself in the process of granting independence to its own Middle Eastern colonies in Jordan and Palestine. In contrast, British interests in Malaya were significantly more permanent in 1946 and aligned to the French. Britain feared the rise of a Communist power in Southeast Asia, at the same time as it was struggling against political (at the time not yet military) unrest fermented by the Malayan Communist Party, which enjoyed broad support and legitimacy due to its actions in WWII (much as the Vietminh did).
In addition, it is important to remember that 1945-1946 was a very different political climate from the years proceeding it. This was still a highly unstable world, and it was thought that Syria/Lebanon/Jordan/Philippines etc., as countries which had already been preparing for independence since before the war, would be less disruptive than new and emboldened states like Vietnam. There was a desire to prevent the emergence of radical states, who were legitimised by the failures of the Western powers, rather than in collaboration and tutelage.