Though both were proxy wars of the Cold War, the circumstances of Korea and Vietnam were different with regard to the run up to war, Korea being a relatively sudden, distinct action and Vietnam being a gradual escalation. For those not familiar with the circumstances of the two conflicts, let’s go to a history lesson for dummies (myself included).
Korea (having been occupied by Japan since the early part of the 20th century) was divided along the 38th parallel following the end of WW2 in 1945 into separate zones of occupation with the Soviet Union responsible for the north and the US for the south. Each enabled the establishment of a government in their respective halves that was favourable to their political ideology. The Allies (UN?) had as their intention the eventual reunification of Korea but no timetable or process had been identified. Following withdrawal of American and Soviet occupation forces from their zones the two Korean governments competed in their attempts to re-unite the peninsula under their own systems. There were limited military attacks along the border through 1949 and early 1950. North Korea, having been more heavily armed by the Soviet Union than South Korea by the Americans, changed the nature of the conflict with a full-scale invasion of the south commencing on 25 June 1950. This was the defining act of overt aggression that was used to request and receive authorization for UN military action.
The story is much more complicated and long-term in regards South Vietnam. At the end of WW2, France attempted to reassert itself in its colonial empire in Indochina (a federation of protectorates, Tonkin, Annam, Cambodia, Laos and one directly ruled colony, Cochinchina). Vietnamese nationalists (the Viet Minh, predominantly communist) opposed them in this led by Ho Chi Minh. For the first few years it was a low-level insurgency against the French authority. After the communists took control in China, they provided direct support to the Viet Minh and it became a more conventional war. This, the First Indochina War, lasted from 1946 to 1954 and was costly to the French in terms of money, casualties, power and prestige. Following the defeat of the French at Dien Bien Phu in 1954, an accord was reached at the Geneva Conference that resulted in independence being given to the former French Indochina possessions. The military action in Vietnam during this period could be viewed as similar in aim to the American Revolution. Vietnam (Tonkin, Annam and Cochinchina) was partitioned into northern and southern zones pending unification on the basis of internationally supervised free elections to be held in July 1956. As an aside, Canada was one of the three countries that made up the International Commission for Supervision and Control that was to monitor the partition process.
During the period 1954 to 1959 the governments of South Vietnam and North Vietnam ruthlessly suppressed political opposition in their own countries. The elections scheduled for July 1956 were not held primarily due to the decision of the South Vietnam government (supported by the USA) to not participate because any election would be unfairly won by the communists. By 1959 the north had reached the conclusion that the country would not be reunited by purely political means and ramped up the campaign of violence and insurgency in the south. The United States had been providing military assistance to the French during their war and continued their involvement in the region by aiding the government of South Vietnam. As the level of violence and insurgency increased so did the military assistance and level of participation from the USA. By 1965 the situation had developed to the point that the USA committed ground combat forces. Over the next few years US troop levels increased as the war escalated.