This is a very reductionist comparition: communism is not only Marxism-Leninism, and socialism is not (modern) social-democracy.

The Encyclopedia Britannica do a better (but incomplet) work:

Exactly how communism differs from socialism has long been a matter of debate. Karl Marx used the terms interchangeably. For many, however, the difference can be seen in the two phases of communism as outlined by Marx. The first is a transitional system in which the working class controls the government and economy yet still pays people according to how long, hard, or well they work. Capitalism and private property exist, though to a limited degree. This phase is widely regarded as socialism. However, in Marx’s fully realized communism, society has no class divisions or government or personal property. The production and distribution of goods is based upon the principle “From each according to his ability, to each according to his needs.”

Or the wikipedia definitions:

Socialism is an economic and political philosophy encompassing diverse economic and social systems characterised by social ownership of the means of production, as opposed to private ownership.

Communism is a sociopolitical, philosophical, and economic ideology within the socialist movement, whose goal is the creation of a communist society, a socioeconomic order centered around common ownership of the means of production, distribution, and exchange that allocates products to everyone in society based on need.