Answer:
Communism has an economy that is completely controlled by the state, while Capitalism has an economy that is almost completely free from state control.
Free education and healthcare can be a major advantage of Communism, if implemented correctly.
An example of a disadvantage of a communist state is the Soviet Union. They every element of life was completely controlled by the state. There were no free and truly democratic elections, and purges ran rampant.
The biggest advantage of capitalism is that is supports people making new, non-state controlled businesses, allowing regular people to make a profit off their companies.
The Capitalist system allows unemployment to run rampant, and people have to pay for their education and healthcare. Just like in the US.
The US wanted to promote Capitalism to stop the Soviet Union from politically (and territorialy) dominating the world. The Soviet Union wanted to stop the US from politically dominating the world, and they wanted to politically, and territorialy, dominate the world.
Hope it helps!