How come there are still two Koreas while the other divisions created by the Cold War have ended? Do the people of Korea want the division to continue?
- The government of the Democratic People's Republic of Korea (DPRK, North Korea) wants reunification on their terms, i.e. the rule of the Kim dynasty over the entire peninsula under the Juche system. The people of the People's Republic are not consulted.
- The people and government of the Republic of Korea (ROK, South Korea) mostly want reunification on their terms, i.e. a multi-party parliamentary system for the entire peninsula. Some might have doubt about extending capitalism and democracy too fast, some might put unification over democracy.
- The DPRK did not collapse the way the GDR (East Germany) did when Soviet/Warsaw Pact/COMECON system collapsed. Part of that was due to a lesser integration into the Soviet system, another part was that the DPRK government decided to hold out. The result was the economic downturn and famine called the Arduous March.
The GDR government might have ordered their troops to shoot at the citizens. The troops might have followed that order. They didn't give the order and the Berlin Wall collapsed.
First it a bit incorrect to call Korea the only divisions created by the Cold War still existing, an other example would be China and Taiwan.
To understand why there is a difference to other examples, we need to look how other divisions have ended. All other divisions have ended with some kind of annexation of one part by an other. While in Germany the annexation was peaceful, so some would disagree with using this term, in Vietnam it was really bloody. The Vietnam solution hasn´t work in Korea as none of the sites was able to conquer another by force. For the solution like in Germany there need to be a regime collapse in one of the parts and even such collapse is not a guaranty for reunification as the regime in South Korea has already collapsed in the 80s but has not lead to reunification.
Sovereign states have a tendency to keep going - once a country exists and is recognized, it is becoming full of its own elites, government officials, state intellectuals - some of whom risk losing their status in case of a merger with another country, which has a set of their own.
This does not fully explain why e.g. corporations are much more eager to merge than countries. But the observation is there.
North Koreans realistically wanted an "one country, two systems" merge with South Korea but it was not possible either politically or economically. They have a border with a largest difference of wealth in a whole world, I believe. And Kim family who want to keep their kingdom really bad.
Since the situation in Korea is often compared to the one that was in Germany, some diferences should be shown:
Germany has a history of being a sovereign state even an empire and a strong national identity, nothing like this was existing in Korea.
From the very begining West part of Germany was more atractive then the East one. At least the first 30 years South Korea was a dictature not any beter then North Korea.
Part of the West Germany (West Berlin) was inside East Germany and until 1961 people from the East were able to see the situation in the West.
In North Korea there is a unike government form, which can be called communistic monarchy where the Kim family has all power. In GDR the first leader was overthrown in an unbloody coup and another similar coup was one of the reasons for the end of GDR. Such coup is very unlikely in North Korea.
An important sorce of power in GDR were the soviet troops. After the USSR refused to protect the government of GDR it was over. North Korea has not to rely on any foreign troops.
As stated in one of the other answers, both Koreas are officially at war. An attempt to reunite militarily, both from one side and the other, is likely to lead to a nuclear war, and this even without the intervention of Russia and China on the side of North Korea.
If consider the possibility of peaceful reunification, then the most likely, and at the moment the only possible option is the annexation of North Korea to South Korea. The anonymous answer contains many differences between the situation with Germany and Korea and I will not repeat them, but some things were not mentioned. Both Koreas are approximately the same in area and population, the preponderance is only 2 times in favor of the south. The eastern part of Germany was several times smaller than the western one both in area and population. Already based on these parameters, the example with Germany does not quite fit. However, certain conclusions were drawn from the example of Germany that are not in favor of reunification. In such a scenario of reunification, the North Korean government will lose power, as well as fall under judicial reprisals, which is definitely not in the interests of this government. Russia and China, which are neighbors of North Korea, are not interested in the appearance of pro-American state or even American troops and weapons on their borders. Although Japan itself is pro-American, it is also not particularly interested in the appearance of a large and united Korea in the neighborhood, which has territorial disputes with Japan. And in South Korea itself, there are opponents of reunification, which will hit the country economically and most likely the pockets of citizens, who, as the example of Germany shows, will have to pay a reunification tax.