It seems that the South Korean government initially had a lot of the same weaknesses that the South Vietnamese one did. Why was the US capable of ensuring the former's survival but not the latter's?
In a strict sense, the South Korean government fell multiple times, to be replaced by new South Korean governments (some more democratic, some less). But I assume your question is about why they didn't fall to the North.
Two big reasons: the Korean War and economic growth.
The Korean War was the North's attempt to unify the country by force. This came within sight of success as the South's forces (including their allies) were almost pushed off the peninsula (but held at the Busan Perimeter). The South's forces pushed back, and made their attempt, in turn, to unite the country by force. This was stopped by the Chinese entry into the war, and the war continued for two more years as a brutal battle of attrition on a front line close to the border, with the USA and South Korea as the principals on one side, and China and North Korea on the other. That front line is still there - there is no peace treaty, and the war is still officially on (but the armistice stopped almost all of the fighting). The war was very destructive, with the front lines crossing almost the whole country twice.
North Korean attacks on the South did continue. Also Southern attacks on the North. These were minor (compared to the real war) border incidents or infiltration missions by small numbers of agents/soldiers - the most famous of these was the North Korean attempt to assassinate the President of South Korea in 1968 which got within 100 metres of the President's residence in Seoul. Any attempt to cross the border in force, to conquer the South by conventional military force, would have just led to re-opening of active full-scale war, and the North's chances of success were essentially zero - the South was now well-equipped, and supported by US forces already in place (and the US forces were better-equipped and much better motivated and trained than the first US forces thrown into the Korean war).
The North Koreans did create a revolutionary movement in the South (Revolutionary Party for Reunification). It was never big, never a major popular movement, and was essentially crushed by the South Korean security apparatus. In this case, repression worked (it does have a modern descendant, now operating as a conventional political party, the Anti-Imperialist National Democratic Front (AINDF)).
So, the North had no real chance to overthrow/conquer the South after their failure to do so in the Korean War. Would a Southern movement overthrow the South? This is essentially what happened a few times. But each time, the new government was a new non-communist Southern government.
The South Korean government was initially formed by people who had been active in the independence movement against the Japanese. For example, the first President, Syngman Rhee, had been Prime Minister of the provisional government established in Shanghai. He had the political status and experience, and US support, to rise to the top as the first President of South Korea. He was brutally repressive, having political opponents killed (including outright massacres, but also including imprisonment and execution of politicians in the opposition). Civilians were being massacred on the basis of being supposedly communist, even before the Korean War, and the Korean War provided more opportunity to have opponents shot (claims of the numbers killed in such massacres during the war vary a lot - the official count for the Bodo League Massacre in 1950 is approximately 5000, but there are non-ridiculous claims of 50-200,000 deaths).
Rhee was also corrupt. This led to difficulties, as the President was elected by the National Assembly (i.e., parliament/congress), who were unlikely to elect him again (on the grounds of his excessive corruption). Rhee's solution: arrest his opponents, and force a constitutional amendment through the now-reduced National Assembly making the President chosen by direct popular vote. This got him through to 1960, when popular reaction to violent repression of popular protest over a too-obviously rigged election led to his fall.
Exit the First Republic, and enter the Second Republic, with reduced powers for the president, and a major role for the Prime Minister. In 1961, Park Chung-hee's coup overthrew the government, and Korea was ruled as a miitary dictatorship (mostly pretending to be a democratic civilian government) until the Sixth Republic, established in 1987. Despite repression during his dictatorship, Park Chung-hee is revered by many Koreans (especially older Koreans) as the architect of Korean economic recovery after the War. With this economic recovery, and the stagnant North Korean economy across the border as the example of what communist rule would do:
there was essentially no chance of any communist movement taking over. Still, "communist" was used as a label to target opponents - for example, the Gwangju Massacre in 1980 - a violent and bloody repression of a protest against continued military rule after the assassination of Park - was described as an attempted communist uprising. One Korean rock star refused to include a song praising the government on his newest album, and was imprisoned and tortured, and his music banned.
Popular protest continued and grew. These were pro-democracy, not communist, but protesters were jailed/tortured as "communists". With frequent protests/riots in Seoul growing in intensity and anger, the government was facing a serious popular uprising. Perhaps because their other choice was to fight to stay in power and be put against the wall and shot when they failed, the government agreed to a peaceful transition to democratic rule, the Sixth Republic. Yet another South Korean Republic had fallen. (And since then, there was the fall of President Park Geun-hye (Park Chung-hee's daughter) through widespread anger over corruption, that was a fall within the existing government structure.)
