When WWII ended , Soviet forces took over the northern half of Korea. While American forces took over the southern half. The two republics divided themselves with the 38th parallel of latitude.
In Korea and Vietnam, the Soviets (and the Communist Chinese) supported Korean and Vietnamese communists living in the northern halves of those countries. The United States supported the more democratic southern regions of those countries.
A similarity between the Vietnam and Korean war is that during the time, they were both fought to contain the spread of communism, both were fought during the cold war, and that they were never an actual declaration of war.
The Vietnam war was officially ended unlike the Korean war. And after the war, Vietnam went back to being a communist country.
The Korean war was more similar to WW2 in terms of fighting tactics, Vietnam was more of a guerrilla warfare
The Korean war, however, was ended in a "cease-fire", meaning the war was never officially ended. After this war, the country divided with the North still being communist and South staying non-communist.
The Korean war was a short one, only lasting three years. But the Vietnam war lasted about twenty years, since it was never officially declared as a war.