Why were the situations in Korea and Cuba so significant in relation to the Cold War competition between USSR and the U.S.?
Since the information Cuba is more available, let me speak on Korea a bit. First, what we are taking about is South Korea, since the US troops are stationed there. In fact, some of my good friends are there now. The significance of this is because during the cold war, China was allied with Russia. North Korea was also allied with them as well. So, in that region, there were very few democratic states that were with the US. To have a base of operation in that area, therefore, was important, even vital. In this sense, South Korea was a holding point against the spread of communism. Of course, things are changing a bit now.
There are collections of reasons to why Korea and Cuba were of such great importance during the Cold War.
Cuba: In 1962, Cold War came very close to erupting into a nuclear war and this crisis was termed the Cuban Missile Crisis in the United States. For USSR, Cuba was an ideal location to place intermediate range missile to intimidate the American government. American missiles were positioned to be able to reach the Soviet lands; however, USSR did not have the technology or a secure position to have their missile reach the United States. USSR's decision to place nuclear warheads came as a response to the United States's placement of two missiles, Jupiter and Thor, in Turkey. Cold War was a game of intimidation and fear; therefore, for USSR, having control over Cuba meant an ultimate threat to the United States's national security.
Korea: One of the biggest concerns during the Cold War between the USSR and the United States was expanding their sphere of influence around the globe. Furthermore, this was one of the main reasons to the development of conflicts in Afghanistan and Vietnam. After the Japanese occupation, USSR got a hold of the North and the United States got a hold of the South and during the 5 year recuperation period for Korea, two different political ideologies deeply rooted itself in Korea. To make it simple, for the USSR, North Korea was a type of buffer zone in the East; for the United States, South Korea was another "outpost" to keep the Soviet in check.