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The spelling is the only difference.

Pretty much. I usually put it this way:
- Under fascism the government tells you what to make in your business, how much to charge for it, who to employ and how much to pay them, who your suppliers will be, and anything else (logos, advertising) that it takes to support the ends of the government. But you have a piece of paper that says you "own" your business.
- Under socialism the government takes the piece of paper as well.
- Under communism they tear up the piece of paper. (Note: There has never been a communist government. The old USSR was the "Union of Soviet Socialist Republics. All the so-called communist regimes are guided by the communist party, but they never quite get around to relinquishing government control of everything.)