No doubt you will get a very scholarly reply from Yoshie and Carrol.
But, I think, off the cuff, most people will say that socialism is "demoratic" communism, while communism is communism led by the dictatorship of the proletariat. How that proletarian dictatorship is organized: in soviets, or a polit bureau, or a single uber leader -- I guess varies with circumstance. I think even with communism, the dictatorship of the proletariat, is meant to be transitional, giving way to a democracy at the point where class identities have been erased by lived reality .
But this is the uninformed definition.
Joanna