What were the differences and similarities in the methods of the wars in Korea and Vietnam?
These two wars are much more different than they are similar in terms of the methods used to fight them. The only real similarity is that the US used as much technology as it had (short of nuclear weapons) to fight the two wars. It generally tried to win the wars through technology.
But there is a huge difference between the two wars. Vietnam was a war fought mainly against guerrillas. It was also a war that was fought within an officially friendly country. Therefore, it was a war that had both a military aspect and an aspect in which the US was trying to win the "hearts and minds" of the people of South Vietnam.
By contrast, Korea was a traditional and conventional war. It was a war in which the two sides were trying to conquer territory. It was a war in which both sides used regular armies to try to push the other army out of a given area. This was a war like WWII, not a guerrilla war like Vietnam.