There was a societal effect too. The country watched, over 13 days, as the whole nightmare played out, and the possibilities of nuclear annihilation were talked about openly on the evening news. In other words, the American and Soviet people looked over the edge of the cliff, and it terrified them, rightly so.
Secondly, the way in which he handled the Cuban Missile Crisis, stared down the Soviets and in the end, protected the country made JFK a much more popular President. He was young, the youngest ever elected, and was criticized by many as being inexperienced, but the Cuban Missile Crisis vetted him as a leader.
In my opinion, the most important effect of the Cuban Missile Crisis was to make the Soviet Union and the United States realize that they needed to have a way to cool tensions with one another in case of another crisis like this one.
During the crisis, there was no real way for the leaders of the two countries to contact one another easily. Therefore, they had to go through middlemen and their messages could be distorted. Because of the crisis, a telephone hotline was set up that would allow the leaders to contact one another easily.
In addition, I think it just made the two sides more aware of how easily things could blow up into a general (nuclear) war and so they started to take their actions more seriously after that, trying hard to avoid anything that would be such a direct confrontation.
Finally, you can argue that the Soviet leadership felt Khrushchev had lost this confrontation and that this feeling led to him being ousted from leadership soon afterwards.