Korea was split at the end of WW2 between the Communist dominated North, and the Western dominated South. South Korea eventually became a shining example of Western style liberal democracy and capitalism. The North remains a "communist" military dictatorship. Vietnam was completely conquered by their northern faction. Korea remains divided by the most heavily guarded border on Earth. I recommend that you look into the differences between the North and South and you'll understand why they are still hostile to each other. Technically they're still at war.

Where do you get the idea that they DON'T want to be unified?

First, the official policy of both governments is reunification. They issued a statement in the 70's giving up the idea of reunification by force and working to improve relations. Here is some information from the South's Unification Ministry on this.

The reality of the situation is that both sides want reunification under their terms. Article 3 of the South Korean constitution claims the entire peninsula as their territory. Article 103 of the North Korean consitution claims Seoul as the capital. Both sides are clear about who they believe is in charge.

The last President of South Korea, Lee Myung-bak, talked about a "reunification tax" to help fund reunification. The current president, Park Geun-hye, claims that reunification will be a "jackpot" (대박) and is pushing for her own funding agenda for reunification.

On the ground, support for reunification has fallen over the years. Reasons for this are the perceived cost involved, the fear of North Koreans coming to the South and ruining the economy, and just a general lack of enthusiasm from the younger generation. Source

Why is it not happening? Neither government is willing to allow the other to be in charge. The South is happy with their lives and the North knows that they'd all be executed/jailed for what they've done. The idea of a confederation has been floated, but any opening of the North results in more dissidents, which they cannot allow.

TLDR - officially, both North and South want reunification. While the numbers have fallen, a majority in the South still want reunification.