The Korean War ended with an armistice, and thus, by technicality never really ended, not that it started "officially" for the US either. In the first phase of the war the Korean Peninsula was almost united twice. Once by the communists, and the Pusan Perimeter, and once by the UN forces right at the Yalow River. After the Chinese got involved, the lines went back to Seoul, then back to the 38th parallel, essentially where the war started. Unlike many of the Cold War's battles, it never had a satisfactory conclusion.
Much like the Berlin Wall, the DMZ in Korea was a scar across the land. It was a division across the land that divided once united peoples. Much like East and West Germany, once the walls went up, there was a solidification of the differences. When the Walls finally fell in Berlin, the East was granted access to the West, and all the benefits of a western society, full of on demand manufacture, a working power grid, and freedom from Soviet style "intelligence" services. The West got a mess it took over 10 years to fix.
There was also another legacy of the Berlin Wall. Once it fell, there was no way that communism in Europe could continue. No way for them to keep out the cultural and economic pressures that the West was so good at applying. Indeed the Soviets were already collapsing from the weight of their inadequate methods of manufacture (Ugo anyone?) their abysmal agricultural practices, and their social decay. The major question seems to be "why did it last so long?"
It has more to do with the fact that it was a stability factor. It prevented the Soviets from being Drastic and gave the West an easily defensible line of embarkation. The Soviets had to cross this line to start the War in Europe. The major powers of America and the USSR put pressure on their local client states to keep these divisions, because it was felt that it gave stability. If Germany had Unified in say 1960, what do you think would happen? the crackdowns nearly did force US involvement, and with Kennedy in charge, how effectual the Military would have been in such an intervention is highly debatable.
Well the cold war is long over. The Belligerent Powers have made peace, and Communism is actually rather like a joke on most of the world stage. Why is Korea still divided? You'd think a whole unified Korea would be good (eventually) for everyone. In the South, their embrace of Capitalism has lead to them to be an industrial, and economic powerhouse. They rival much of their own regional adversaries, like Japan and China. In the North there is Famine, and Stagnation. Both of which can be seen in the fact that North Koreans are typically several centimeters shorter than their southern counterparts despite being racially identical. One would think that (aside from those in power) the North would be clamoring for the benefits of the South. But that wall is not going to come down. Much like the Berlin Wall the regional (and world powers involved) don't want it to happen, but also unlike Berlin, the people inside Korea don't want it to happen.
To understand the outside influence, you have to understand that the North shares a Border with China and Russia. The Russians do not want to lose access to their ports, in that area because really that's one of the few port facilities Russia has access to that don't freeze up in winter. The Chinese do not like rivals. The Vietnamese and Chinese fought a terribly bloody war, after the US fought there, and the Chinese always are a little paranoid about invaders. They're paranoid about the nationalist Chinese in Taiwan, which, despite being given hardly any land at all and hardly any resources are constantly elbowing China in important markets, and most often doing the same jobs better quicker and more efficiently. If North Korea, which is heavily dependent on Chinese good will were to become as prosperous as the South, and indeed Unified, it is entirely possible that they might be forced out of many of the markets China depends on (in their own little way they did embrace capitalism in the end) and worse they wouldn't have one of their key playing cards against the US and the West.
In the "Free World" you have Japan, a long time rival and enemy of Korea. Pretty much every time the Japanese got sick of their little islands or wanted new land the first place they went was Korea. This explains why there is so many ethnic Koreans in Japan, they were often brought back and used for cheap labor, and with Japans inbuilt racial stresses, such gaijin are never allowed the benefits of citizenship. History of conflict between Korea and Japan is long, bloody, and probably too dull for most Westerners to muddle through. Indeed Korea would not even be divided if not for Japan. Japan fears the hatred they engender in the Koreans (rightly so) and a Unified Korea would not only be an economic threat (Japan's economy is not the best in the world) but a military one as well. The US, would be fine with it, so long as it did not cause regional chaos, though knowing our more Progressive "friends" they would preserve the status quo until Judgement Day
But wouldn't the Koreans want it? I point back to Berlin. It took 10 years to make the East as in anyway comparable to the West. How long do you think it would take North Korea, which has been run Piss Poor for decades, to recover? Never mind the constant animosity, of the North sending their special forces on "training missions" to kill South Koreans. The rumors of millions of artillery rounds being pointed at Seoul for over 60 years has not helped much. The anger and bitterness caused by the division would probably take almost a whole generation to heal.
With the Kim Dynasty passing on to the third generation, there were (initially) many people in high places that were worried that the relatively unknown Kim Jong Un would be unable to control his country. It is still possible that there could be chaos. If Kim Jong Un can not control his military, or perhaps worse, if he can only control a portion of his military, there is a potential for collapse, and civil war. A war which would *naturally* draw in most of the "belligerents" that have kept Korea divided. Throw in Nuclear Weapons in the mix and you reach a conclusion that many do not want to contemplate, which is why, many in the halls of power continue to hope that the Wall will stay up, and why no one has bothered to plan for the day when it *eventually* will fall.