Actually socialism is older than communism.
Prior to Marx, there was "Utopian Socialism". Marx found those ideas unrealistic and critiqued them.
Basically after Marx, all socialists came around to a more Marxist way of looking at things.

The main textbook definition is that communism is supposed to come about just like Marx said... through evolution of the economy, the spread of capitalism, problems of capitalism becoming obvious, and the workers ultimately leading a revolution against the owners.
Socialists apply the critiques of capitalism but choose to do it without revolution by working within the system.

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"It's never over... BOY!"
The Tall Man, Phantasm III