Because "North Korea" and "South Korea" are different countries. Normally when you say "Korea" in English, it means South Korea. So if you said "Northern Korea", I would think you mean the upper part of South Korea. Another example: "South Africa" = a country, but "southern Africa" = the southern part of the continent. (note: no capital "S")

The shorter form is usually preferred for any geographical name where the boundaries are clear, either because they are officially defined or generally recognised. That's not just for Korea, but for any other country or region, too, e.g. North America, South Africa, East Timor and so on. "Northern", etc. are used to describe more a more general or context-specific side of a region. So for example, we might talk about "southern Africa" and include such places as Namibia or Botswana in our meaning, rather than just the country of South Africa.