The terms themselves are somewhat vague and slippery. Marx and Engels used them interchangeably. The USSR and China really tainted the word communism, which is why socialism is much more common nowadays.

As I understand it, communism is a form of socialism. Socialism is ultimately about worker control over the means of production, rather than private capital. As such, socialists inherently support strong unions, and the sensible ones also support social welfare, minimum wage, and basic income so that business owners have less leverage to exploit their workers.

If you just take workers’ rights to it logical conclusio, you get market socialism. This is an economic system in which all privately-owned (including publicly-traded) companies are replaced with worker-owned coöperatives, which still compete in a market.

Communism goes further. Self-identified communists will tell you that communism is a moneyless. classless, stateless society where the means of production are held in common by those who use them. If this sounds like anarchism, it basically is.

However, communists in the 20th century were mostly vamguardists. This idea, pioneered by Lenin, advocates for a vanguard of smarties who understand communism to overthrow the government and impose communism from the top down, fixing the system on behalf of those workers too stupid to join the revolution. Workers who did not support the revolution would see that everything was much better with the communist vanguard in charge, and would embrace communism. If a few insisted on being counterrevolutionary, they would just need to be reëducated.

The Russian Revolution was heavily criticised by anarchists at the time, on the grounds that if the revolution does not rise from below, it is simply a coup that makes Lenin an uncrowned tsar. They were correct, and thus the word communism was utterly tainted in the capitalist world to refer to oppressive dictatotships that are (nominally) anticapitalist.

For what it’s worth, Lenin himself described the USSR as state capitalist, whereby the state ran all industry on behalf of the workers until the workers came around to the glorious revolutionaries’ perspective. Because those in.power never want to relinquish it, the ruling soviet aggressively cracked down on and suppressed trade unions, because organised workers were a threat not only to capitalists, but also to the nominally communist government. To maintain a veneer of being about the workers, farms and factories were administered by soviets vetted and approved by the government, who could be guaranteed to operate as the government wanted.