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The summit itself had a longer prehistory and is deeply tied to a number of other events; it is projection from hindsight to connect it directly to the Soviet stationing of missiles in Cuba, though of course it was eventually a contributing factor to this as well.

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1. The General Strategic Situation

While Kennedy had won his 1960 election on the notion that the Soviet Union was ahead in the arms race and that a "missile gap" had developed the West had to allegedly catch up with, it was (among the more reasonable Pentagon estimates) actually assumed that this was not the case, but it was assumed that the Soviets were catching up fast. The problem lay not so much in the question of intercontinental missiles (that could reach the United States from the Soviet Union) but in short- and mid-range nuclear weaponry, in which the Soviets were building up an increasingly significant deterrent beginning to match the combined US-UK deterrent; in ICBMs, however, the Soviets were lagging, and the RAND corporation calculated that a preponderance of roughly 6 to 1 could be enough for even a reasonably successful first strike leaving the United States largely unscathed in a (theorized) nuclear exchange.

In ICBMs, the United States did have that preponderance, but it was unclear whether that was the case for the shorter range weapons. The Soviet buildup was therefore fast leading to a situation where the United States did not have a "reasonable" (i.e. mathematical) chance at a successful first strike if on a full global scale, thus rendering Soviet conventional preponderance vs NATO in Europe (larger number of tanks and divisions) more meaningful. It was this basic outlook that drove a certain desire in the defense community to look for "other" solutions, like arms limitation treaties, combined with diplomacy and the rise of the "flexible response" military philosophy. The Kennedy administration had decided to switch into this "newer" deterrence philosophy, which required a position of strength from which to negotiate such an outcome and preserve the status quo and a general preponderance of the West as a whole. This explains the general raise of the nuclear defense budget, politically supported by the (knowingly false) missile gap rhetoric, which was meant to garner cross-party and public support for a raise of the military budget.

This general strategic concept transition, however, was basically interfered with by the previous immediate concrete political events both sides of the Cold War were engaged in. These were, in particular Laos, Berlin, and Cuba.

Soviet premier Nikita Khrushchev, who was acutely aware of the actual Soviet weakness vis a vis US nuclear power, needed to beef up the image of Soviet capabilities to force the United States to the negotiation table, while hectically trying to speed up the Soviet nuclear arms buildup. This played into the hands of those in the United States who were looking for larger defense budgets, and were eager to use Khrushchev's various boisterous claims for US domestic purposes of amassing political support for ever bigger arms budgets.

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2. The diplomatic prehistory to the summit

Pre-election "feelers" sent out by the Kennedy team to Khrushchev as early as November 1960 were asking the Soviet side to agree to a summit in the "spring", hoping in return for a Soviet "quiet" period in which the Soviets would not exacerbate any situation in the crisis hot spots until Kennedy had met with Khrushchev. There was a faint hope that this might be the beginning of arms limitation talks and even disarmament agreements, but... politics intervened. Offering a Soviet leader a summit was, even though Eisenhower had done this before, still a pretty big boon to offer; the previous summit in Paris, however, had only served the West to "play for time" with Khrushchev, which had therefore ended with the Soviets breaking off the summit (cf. U2 espionage flights). Khrushchev had hoped that some form of movement could begin on the Berlin question, which the West tried to avoid at all cost: Berlin had opened several cracks within the Western alliance, as neither France, the UK, the US nor Bonn (West Germany) could fully agree how to deal with the situation there. Nevertheless, at first the internal papers of the Kennedy White House show a certain optimism to get to terms with Khrushchev by direct bilateral talks.
Khrushchev indeed responded at first by adhering to the seeming US signals: there was no immediate repetition of threats or ultimata in the first few months of 1961. But Kennedy had inherited a fateful covert ops plan from the Eisenhower administration: the plan to overthrow Fidel Castro's regime in Cuba by landing CIA-trained armed Cuban exiles there and begin a "rollback" of Communist control over the island. Kennedy was wary that this plan might backfire elsewhere, but had committed to it insofar as the removal of Castro was considered a desirable goal. However, he did draw the line at the involvement of US troops directly. Changes to the plan were made that would limit the "visibility" of US involvement, as the White House refused to engage in an action that would draw the United States into a direct open war against a Soviet ally. That, however, had been clearly the CIA plan's intention. When the plan had changed to land during a bright full moon night at a fairly easily defendable beach, it was pretty much doomed to fail, and it is fair to say the CIA let the president walk into this mess without giving good advice and totally overestimating their own capabilities. The Bay of Pigs landing is widely and rightly considered a military disaster, and there is enough evidence to suggest that a) the Soviets had caught wind of the plan in time, had b) warned Castro about it (who famously had his troops ready near the beach within hours of the landing), and, worst, c) the CIA had received warning that the KGB might have gotten hold of the plan and did not inform the President.

3. The Bay of Pigs changes the nature of the summit

After the major public disaster at the Bay of Pigs, the White House was under severe public pressure; to his domestic opponents, Kennedy now looked weak. To the Soviets, he looked indecisive while simultaneously they had been publicly "slapped in the face" by having an ally of theirs attacked while, from the Kremlin's perspective, they had hold their horses still to begin talks with Kennedy at the already agreed summit in June. Khrushchev's response was immediate: in a private letter to Kennedy (published today in the Foreign Relations of the United States series, but top secret at the time), Khrushchev warned that (I'm paraphrasing) "setting fire in one part of the world will kindle flames elsewhere" -- which was a barely veiled threat against West Berlin. Kennedy immediately responded by asking for calm, but the damage was done: both sides, Moscow and Washington, immediately began preparing for a summit that would be a stare-down "test of nerves", rather than a serious sitting down to talk about world problems. The White House immediately began drafting even further increase requests for the military budget, and Khrushchev resumed threatening a unilateral solution to the "Berlin problem".

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4. The Summit is not actually much about Cuba
The three day summit in Vienna consisted of two days of talks between Kennedy and Khrushchev with various foreign secretaries and experts present at round tables, while only the third day had Kennedy speak to Khrushchev alone with only interpreters present. Talks were held on Laos, Cuba and - most important to Khrushchev - Berlin. It was during this meeting that Khrushchev, even when alone with Kennedy, reiterated that foremost the Berlin question had to be settled by the end of the year. While that was a repetition of an ultimatum Khrushchev had made twice before (58 and 60), it was clear that the Bay of Pigs had indeed worsened the Cold War considerably, while not earning Washington any favors with anybody. In the first days after the summit, most journalists presented a picture of a bullying Khrushchev harassing an overly idealistic naive US president. But the truth was that Kennedy had held up to Khrushchev signaling the United States would very definitely not back off in Berlin (and that on Laos, agreement could perhaps be found). Nevertheless, the White House was hoping Khrushchev would not reiterate his ultimatum on Berlin publicly, so that the crisis could "fade away". But the Soviets chose to publish the summit meeting minutes within a fortnight after, repeating that Western allied military presence in West Berlin would come to an end "within the year": the second Berlin Crisis was now coming to a head.
It was this move that propelled the meaning of the summit into something else: what was originally intended to be a serious sitdown between the leaders of the two world camps had first deteriorated into a standoff show of strength, and then became a platform for the initiation of the fall 1961 Berlin Crisis.
There is still considerable debate about how these different set pieces of the cold war - Berlin, Indochina and Cuba - were precisely linked, but in the case of Vienna it is fair to judge that the United States had misread Cuba as a "domestic" issue, not realizing that to the Soviet Union it had become "their" Berlin issue. The Kennedy administrations continuing attempts to dislodge Fidel Castro (assassination attempts, Operation Mongoose, etc) exacerbated the conflict between two rivaling superpowers, peaking with the Cuban Missile Crisis a year later (which, interestingly, triggered Kennedy's question to his NSC staff "Is this actually about Berlin?").
For a longer and in-depth discussion of these connections I recommend
Fabian Rueger, Kennedy, Adenauer and the Making of the Berlin Wall, 1958-1961, Dissertation, Stanford University (2011) [pdf download here].
which has also all the relevant footnotes.

5. The Vienna summit loses relevance after the August-October 1961 Berlin Crises
The summer months immediately after the Vienna summit saw a major military buildup: As Khrushchev reiterated his threats against Berlin, the Kennedy administration launched into signaling a major war danger to the Soviets by sending large contingents of US fighter and bomber planes to Europe (the largest since World War II), calling up reserve divisions and Kennedy giving a July 25 TV address to the American people declaring the United States would fight for West Berlin and comparing it to Bastogne (a French city successfully held in '44 by American troops against an almost overwhelming German assault). The Soviets had decided to let the East German government build a wall around West Berlin, and there is enough evidence to suggest the Kennedy administration had signalled that a wall was not a casus belli, but an invasion of West Berlin was. This was emphasised further by the Checkpoint Charlie Tank standoff in Berlin in October, where the East German and Soviet governments learned that the West would not budge any further.
I dare say that by the spring of '62, while the Berlin crisis was not completely over, the summit had lost all relevance: the focus of the cold war then moved on to other areas, as at least in Europe the status quo had been pretty much preserved without both sides losing face, while the Kennedy administrations' desire to remove Fidel Castro was still apparent, and Khrushchev had in turn still something to lose there.

On a final note, it is often overlooked by people who dream up alternative histories that if the Bay of Pigs had been used to foment a full scale US invasion of Cuba, the results would have been disastrous: the Soviet Union had at this point already installed nuclear grenades in Cuba (but not missiles), which only became known after the end of the Cold War. This would have easily turned any major naval landing of American forces in Cuba into an absolute nightmare. It is thus fair to say that Kennedy did the absolute right thing to back off any CIA attempts to drag him into an American invasion of Communist Cuba.