Because the Korean War was never actually "settled."

The text of the armistice reads, in part: (bolding mine)

The undersigned, the Commander-in-Chief, United Nations Command, on the one hand, and the Supreme Commander of the Korean People's Army and the Commander of the Chinese People's Volunteers, on the other hand, in the interest of stopping the Korean conflict, with its great toil of suffering and bloodshed on both sides, and with the objective of establishing an armistice which will insure a complete cessation of hostilities and of all acts of armed force in Korea until a final peaceful settlement is achieved, do individually, collectively, and mutually agree to accept and to be bound and governed by the conditions and terms of armistice set forth in the following articles and paragraphs, which said conditions and terms are intended to be purely military in character and to pertain solely to the belligerents in Korea:

First, you'll notice that the Republic of Korea is NOT mentioned here and no one from the ROK signed the armistice. It was signed by Nam Il, the DPRK general, and William K. Harrison, a US Army Lt. General. Syngman Rhee wanted to continue the war which is, in part, why the Korean War ended in 1953 instead of 1951. By mid-1951, the war was a stalemate and peace talks had begun. The Communist forces wanted all POW's to be repatriated, even if they didn't want to go back. Many of the Chinese POW's had been nationalist soldiers and preferred to go to Taiwan while many North Korean soldiers were either not Communist or were from the South and had been pressed in to service. By 1953, both sides had come to an agreement and Operation Little Switch happened. OLS was a trial run where the Communist and UN forces exchanged a few POW's to test everything out. Syngman Rhee was opposed to the whole thing so he decided to let some 25,000 POW and communist sympathizers go in June. Rhee wanted to continue the war and reunify Korea.

Second, the purpose of the armistice was to stop fighting so that a peaceful settlement could be achieved. Remember, the UN had set up elections in 1948 but the Soviets boycotted them and held there own in the North while the US/UN held their own in the South. The war began when Kim Il-sung decided to attempt reunification by force. Had the US armed Syngman Rhee the way that he wanted, he would have started the war earlier. He was just as eager to reunify the country under HIS regime as Kim was to do for the communists.

After the armistice, Operation Big Switch continued until December of 1953. At that point, Rhee and Kim were still in power, they both wanted to reunify under their respective terms, the Chinese and Americans were in no mood to start fighting again (the Korean War was just as much a Communist China assault on the US for their support of the ROC/Chiang Kei-shek as it was for COMINTERN), and both sides were busy rebuilding their country. In the 60's, there were border raids culminating in the Blue House raid of 1968. In the early 70's, the North and South met and issued a declaration in 1972 that said, in part, that reunification must be peaceful and that it must come from within; no foreign interference. There have been a few other joint declarations and discussions, but nothing more has really come of it.

The North Koreans frequently make overtures for a peace treaty with the United States, but the US has routinely told them to work it out with the South.

TLDR: the armistice was a cessation of hostilities so that the sides could start working on a peaceful solution, be that one unified Korea or something else. Due to many different factors, this just hasn't happened yet.