Socialism is an economic system, while Communism has both a Socialist economy and a ttotalitarian state or dictatorship. Communist countries have attempted to have more pure Socialism than non-Communist ones.
In Socialist economies, the government owns many or all of the major industries and operates them as non-profits. Europe and the United States have varying degrees of this. Public schools, bus and subway systems, and the Post Office are socialized in the United States, while some European countries have full socialized medicine, Communist countries have no private ownership of property or factories, and also control prices and the value of their currency, so it is largely an artificial economy, though relatively stable.
Also in Communist countries, it is illegal to be wealthy, which is why so many wealthy Cubans fled from Fidel Castro. They seek to create a "classless" society with no rich and no poor.