Anarchism is a state of being. Communism is a prescription for social inequality.
If one is an anarchist (which is not political, or reactive, or seeking attention, or childish in any way) they are simply inconducive with hierarchy. It is a constitutional divergence from the normal preoccupation (we admire rich people, we are subservient to power centers [human], we adore institutions, etc.).
Communists say, the proletariat should unite and lead society. However, the proletariat must be educated. In a world where innumerable people are uneducated and refuse to educate themselves, Communism remains a distant if not impossible goal.
Socialism, as defined by Marx and Engels in The Communist Manifesto, is led by the bourgeoisie, as a moral predetermination of social equitability; it is a Christianization of capitalism, not an upheaval.
We should remember that the pamphlet known as The Communist Manifesto was released in 1848, on the heels of massive uprisings across Europe, which led to nationalism—not Communism. It led to more power for the bourgeois class, not extending down to the proletariat. Socialism always extends the power of the capitalized middle, never extending to the workers or destitute; the youth or the revolutionaries. Marx was wrong about this expansion.
Anarchism is a state of mind which says, nobody is better than someone else based on any material characteristic. Rich people have their modes of becoming richer, poor people have their modes of manipulating rich people, and people in the middle are often complicit in extending the power of the rich, or the government, or whoever they see as their benefactors which sustain them.
In an Anarchist society, we simply do not adhere to social constraints and prerequisites. We do not see higher and lower based upon financial capacity. We do not recognize 'elites' for any contribution, nor do we look down on those who do nothing for society. We see a chain of being which is ever unbroken, in which everyone is an equal.