It’ll happen. Might be awhile, but the people of South Korea want it.
My understanding is that the older generation of South Koreans, who remember a unified Korea or the aftermath of separation, who remember family across the border, still want reunification
But the younger generation of South Koreans, born into a modernized world of South Korea, and who do not really know family in the North, are not much interested in it at all.
In fact, the chances of unification based on the people's emotions drops every day. As the gap between the two widens, so too does the burden each South Korean must bear to modernize the further-laggard North.
Armistice is a formal agreement to cease all hostilities for likely an indeterminate amount of time to facilitate a peace process. It's making an agreement that War is no longer an extension of Diplomacy, but rather that the parties will use Diplomacy. A ceasefire is an agreement to stop aggressive actions for a predetermined amount of time, calming things down, but has the expectation that war will resume. A truce is an informal agreement , e.g., the famous Christmas Truce of The Great War to a cease fire.
Because the aggression still exists, the reason for the war was never resolved or settled in any way. They just said we both agree not to fight for quite a while, it's formal. It's just lasted a lot longer than an armistice normally does. You usually get into peace talks and sign a peace treaty after an armistice. The Koreas can't agree on terms so they've remained separate. They'll need to eventually find a resolution for a form of peace or else the armistice is likely to fail/end.
For the cynics/pragmatist, there's not much of a difference. You can sign a peace treaty and still violate it or go back to war anyway. Now that may get other nations to rally against you after doing such a thing, but it has never stopped being a possibility.
For all that we know, North Korea and South Korea's peace may take the form of formally becoming separate nations and potentially allowing free travel between the- yea that's not going to happen. peace between them will likely require the hostility to completely end and for South Korean and North Korean citizens to either freely see one another or to officially forsake each other permanently. You'd likely need the North Korean regime toppled or to be made the official new top dogs in the new country.
There are many, many ways this could go. I'm no expert, I just know that a peaceful end to the armistice there is likely not going to be a conventional end.