TL;DR You asked for a short answer and a quick explanation, so I will give you one.
"Communism" is another name for "Marxism", which has its roots in "Socialism," but nevertheless is an offshoot from it.
Here's the longer answer:
Communism is defined by Lexico (Powered by Oxford?) as (emphasis mine):
A political theory derived from Karl Marx, advocating class war and leading to a society in which all property is publicly owned and each person works and is paid according to their abilities and needs.
Therefore, Communism generally referrers to Marxism, that is, abiding by the writing of Karl Marx.
Socialism is defined as:
A political and economic theory of social organization which advocates that the means of production, distribution, and exchange should be owned or regulated by the community as a whole.
Investopedia says the following about Communism:
Communism traces its roots to "The Communist Manifesto," which laid out a theory of history as a struggle between economic classes, which will inevitably come to a head through a violent overthrow of capitalist society, just as feudal society was violently overthrown during the French Revolution, paving the way for bourgeois hegemony (the bourgeoisie is the class that controls the means of economic production).
Following the communist revolution, Marx argued, workers (the proletariat) would take control of the means of production. After a period of transition, the government would fade away, as workers build a classless society and an economy based on common ownership. Production and consumption would reach an equilibrium: "from each according to his ability, to each according to his need." Religion and the family, institutions of social control that were used to subjugate the working class, would go the way of the government and private ownership.
On the other hand, it mentions that Socialism actually pre-dates Communism:
Socialism predates the Communist Manifesto by a few decades. Early versions of socialist thought were articulated by Henri de Saint-Simon (1760–1825), who was himself an admirer of ur-capitalist Adam Smith, but whose followers developed utopian socialism; Robert Owen (1771–1858); Charles Fourier (1772–1837); Pierre Leroux (1797–1871); and Pierre-Joseph Proudhon (1809–1865), who is famous for declaring that "property is theft."7
Marxism emerged in this milieu. Engels called it "scientific socialism" to distinguish it from the "feudal," "petty-bourgeois," "German," "conservative," and "critical-utopian" strains the Communist Manifesto singled out for criticism. Socialism was a diffuse bundle of competing ideologies in its early days, and it stayed that way. Part of the reason is that the first chancellor of newly unified Germany, Otto von Bismarck, stole the socialists' thunder when he implemented a number of their policies. Bismarck was no friend to socialist ideologues, whom he called "enemies of the Reich," but he created the West's first welfare state and implemented universal male suffrage in order to head off the left's ideological challenge.
So it seems that Communism has some roots in Socialism, even being called "scientific socialism."
Wikipedia says the following about socialism:
By the late 19th century, after the work of Karl Marx and his collaborator Friedrich Engels, socialism had come to signify opposition to capitalism and advocacy for a post-capitalist system based on some form of social ownership of the means of production.
Communism, thus, has its roots based in Socialism. Consequently, the two are very similar. That's why the two are often used interchangeably. However, they are not the same.
In general, it might be helpful to think of Socialism as a more general philosophy and Communism as a specific offshoot (Marx's writings) of that philosophy.
As for the practical differences between the two, one source put it this way:
Under communism, the people are compensated or provided for based on their needs. In a pure communist society, the government provides most or all food, clothing, housing and other necessities based on what it considers the needs of the people. Socialism is based on the premise the people will be compensated based on the level of their individual contribution to the economy. Effort and innovation are thus rewarded under socialism.
The same source has a nice table breaking down the differences which is difficult to transcribe here, so I will omit it. But it has some nice differences.
One especially notable one is that in Socialism (emphasis mine),
Individuals own personal property but all industrial and production capacity is communally owned and managed by a democratically elected government.
whereas in Communism,
All economic resources are publicly owned and controlled by the government. Individuals hold no personal property or assets.
Another difference mentioned is in Communism,
Production is intended to meet all basic human needs and is distributed to the people at no charge.
whereas in Socialism,
Production is intended to meet individual and societal needs and distributed according to individual ability and contribution.
Another source notes another difference between Socialism and Communism:
Another key difference between socialism and communism is the means of achieving them. In communism, a violent revolution in which the workers rise up against the middle and upper classes is seen as an inevitable part of achieving a pure communist state. Socialism is a less rigid, more flexible ideology. Its adherents seek change and reform, but insist on making these changes through democratic processes within the existing social and political structure, not overthrowing that structure.
So Communism seems to be associated with a violent revolution, whereas in Socialism this isn't the case.