Both Pres. Eisenhower and Pres. Kennedy were firmly committed to the containment of communism throughout the world. The main difference between them was in their visions of how to accomplish this containment.
Eisenhower's time in office was known for the policy of "massive retaliation" that was put forward by John Foster Dulles. Under this policy, the US reduced its conventional weapons (to save money) and relied on its nuclear arsenal. The idea was that the US would threaten to retaliate in a massive way against any attacker. This would deter attacks without the need for a large and expensive conventional military.
By contrast, JFK's plan was known as "flexible response." JFK believed that massive retaliation did not give him enough ways to respond to relatively small moves (like the communist infiltration of South Vietnam). He couldn't really attack the Soviet Union with nuclear weapons over that, but he did want to try to stop the infiltration. Therefore, Kennedy put much more emphasis on conventional forces, and especially special forces like the Green Berets who were supposed to be able to go in and teach natives how to fight on their own.
So the major difference between the two is that Eisenhower was going to rely on nuclear weapons while JFK pushed for a more flexible set of responses to various threats.