I think you will need someone who is more expert in the Cuban Missile Crisis than myself to give you more information as to the importance of Pope John XXIII's contributions to that event. I can provide a bit more context regarding the pontiff's actions.

For most people reading the news, there was a sense of helplessness before these great events. But at Phillips Academy in Andover, a small conclave assembled that would prove to be highly relevant to the events taking place in Washington, Moscow, and Cuba. The so-called Dartmouth Conference was a gathering of American and Russian thinkers who had met annually since 1960 to improve relations between their countries. As these citizens watched the news, they resolved to do more than simply accept a slow descent into the unthinkable. A visiting priest from the Vatican, Father Felix Morlion, asked the Russians and Americans there if an intervention from the pope might help avert war. One of the Americans, a magazine editor named Norman Cousins, phoned the White House and spoke to Ted Sorensen, President Kennedy’s chief speechwriter. Sorensen confirmed that a papal statement might help, and soon, from Rome, a message went around the world, calling on the leaders to avoid a nuclear holocaust, and reminding them that the fate of all peoples was involved.

Nikita Khrushchev, the leader of the Soviet Union, later called that statement “the only gleam of hope” in a moment of near-total darkness. Russia’s Communist leaders were famously skeptical of religion, ordinarily. But the pope’s message resonated, and provided some much-needed diplomatic cover. Soon after, Khrushchev withdrew the missiles, and the world exhaled. (Trouble in Cuba? Call the pope Boston Globe, Ted Widmer, Jan 11, 2015)

That takes care of the mechanics of the situation. A priest inquired at a conference if papal intervention might help and received a positive response, so the pope made his speech. I can't find the full text, but Vatican Radio gives us a brief excerpt.

In part it read, “I beg heads of state not to remain insensitive to the cry of humanity: peace, peace. Let them do all that is in their power to save peace; in this way they will avoid the horrors of a war, the appalling consequences of which no one could predict. Let them continue to negotiate….”

Pope John’s message appeared in Pravda, the official Communist newspaper, on October 26 under the headline, “We beg all rulers not to be deaf to the cry of humanity.” Khrushchev was given an out. By withdrawing he would be known as a man of peace. Two days later on October 28, Khrushchev agreed to withdraw the missiles.

So there are a couple brief snippets of the speech, including the most famous excerpts. As for analysis of the importance of these words, I cannot give you my own. I can offer historian Ronald Rychlak, a historian who is one of the few that I can recommend regarding John XXIII's predecessor Pius XII. His analysis is here.

Pope John’s role in the resolution of the Cuban missile crisis is often overlooked, but it was very important. It also helped move the world in a positive direction, consistent with the Second Vatican Council that was then taking place. Soviet and American leaders signed a nuclear test ban on July 25, 1963. President Kennedy called that “the first step down the path of peace.” The two nations also set up a “hot line” for emergency messages between Washington and Moscow.

John W. O'Malley S.J. echoes much of Rychlak's sentiments on page 297 of A History of the Popes.

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Now, I can tell you that John XXIII had a different approach to his predecessor Pius XII, especially in how to approach communism. Pius XII was staunchly anticommunist and famously so. John XXIII had no real affection for communism, but his approach was more along the lines of engagement rather than confrontation. The first bit of evidence for that approach is the speech that we have already discussed which was inclusive regarding the Soviets. The instance with more long-term effects was the encyclical Pacem in Terris. That document was addressed "to all Men of Good Will," a key point--and a first. Considering that the pontiff wrote it after being diagnosed with cancer and that he died a mere two months after it was published we should be able to conclude its importance to him. This encyclical was a landmark assertion of human rights, the rights of the individual, the duties of the state, and the need for greater communication and assistance between nations. Nothing was entirely new in the document, but it was given a new emphasis--particularly in the wake of the Cuban Missile Crisis. Such engagement with communist nations was unheard of during his predecessor's pontificate, and the open communication between the Vatican and the Soviet Union was unprecedented as well.

As I said, I cannot directly address your question of the importance of John XXIII's actions during the Cuban Missile Crisis. However, his actions were ground-breaking, and quite different from those of Pius XII. I hope this post sheds some light on the subject even if it is a bit tangential. Followup questions from OP and others are always encouraged.