Exactly as the title says, how relevent was North Korea and South Korea was in between these 2 events and how did it matter internationally?
The reasons shifted over time. America initially got involved in the Korean War to prevent a communist revolutionary (Kim Il-sung) from taking over the peninsula. China, freshly ready to assert its newfound power after resolving its own civil war, intervened with Soviet support.
After the war turned into a stalemate, both America and the USSR/CCCP sent military and economic support to their respective Koreas. For America, South Korea was an important part of their strategy of preventing a wave of communist governments in East Asia. This was the same reason they got involved in Vietnam and Cambodia in the 60s. That status quo persisted for, oh, forty years or so. Then the USSR collapsed, and overnight, North Korea became a completely different type of problem. Before, the problem was "how do we stop communism from spreading into East Asia?" Now, the problem was "what the fuck do we do about this little mini-Mao and his batshit crazy Orwellian nightmare of a regime?"
The answer was "nothing," because when you think about it, North Korea actually just wasn't that important after the USSR collapsed. There was no fearsome "red wave" to hold back; China had opened its economy and was communist in name only; South Korea was turning into an economic powerhouse. Everyone just expected North Korea to implode without support from the USSR. Except it didn't; its economy collapsed and a colossal famine in the late 90s killed millions, but China bailed them out, seeing North Korea still useful as a buffer zone against the US.
But other than that, everybody just ignored North Korea. An intervention was out of the question; North Korea had essentially achieved full deterrence a long time ago, because they had (and still have) several thousand artillery pieces aimed directly at Seoul. And whenever they wanted some concessions internationally, they would take some hostages or sink a ship or shoot down an airplane and bring the peninsula to the brink of war, knowing all the time that any military action against North Korea would result in millions of dead South Koreans and probably the devastation of the growing South Korean economy.
So the DPRK was, in effect, totally safe from outside attack. But they didn't think they were safe. To the Kim regime, the only assurance of complete safety lay in securing nuclear weapons. They knew they could do it; South Africa, India, Pakistan and Israel all acquired nuclear weapons in spite of the NPT. It was just a matter of time and money. The international community was all to eager to continue ignoring them, which solved the "time" part of the equation; the "money" part was hard, considering that they were broke all through the 90s and beyond, but they resorted to selling drugs, sex slaves and pirated CDs (no shit) to supplement government income. Successive attempts to negotiate an end to the North Korean nuclear program were sabotaged largely by North Korea repeatedly breaking any agreement they struck, then starting their brinksmanship cycle again in order to get more concessions in exchange for promises that they would again break. And then, while the world was largely not paying attention, they suddenly detonated a nuclear weapon.
"Oh, they're lying," said everyone. And then they did another one, and another. "Okay, they have nukes, but they can't send them anywhere, surely." A few years later, they're testing ballistic missiles. Here comes 2017, we've all been ignoring North Korea and hoping it would just solve itself for literally twenty-seven years, and they test a working ICBM that can carry a nuclear warhead. Suddenly they're a nuclear power. Now they're a threat. Not just to South Koreans or to US soldiers abroad, but to the mainland - to people in Hawaii and California and even Washington DC. Suddenly, a few million dead South Koreans doesn't seem as unthinkable a price as it was before.
The only hope anyone has of avoiding a dangerous new status quo, where North Korea sells nukes to terrorists to pay for its army and engages in ever-more-deadly games of nuclear brinkmanship, lies in Donald Trump. That, my friends, keeps me awake at night.