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A few things draw significant differences.

Anarchism is fundamentally a firm rejection of unjust hierarchy, including the state, via building up of bottom-up structures using networks of Mutual Aid or other strategies (like Syndicalism).

Communism is fundamentally about advancing beyond Capitalism into Socialism and eventually Communism. It's fundamentally Marxist, unlike most forms of Anarchism (which don't necessarily reject Marx, but also don't accept everything Marx wrote). Communists are generally perfectly fine with using the state in order to eventually achieve a Stateless, Classless, Moneyless society, as each becomes unnecessary and whithers away.

In essence, Anarchism rejects that a state is necessary at all, and seeks to directly replace current systems with the end-goal of an Anarchist structure, whereas Communists tend to agree more with gradual change, rapidly building up the productive forces, and achieving a global, international Communism.

Anarcho-Communism seeks to combine these into directly implementing full Communism without going through Socialism first.

All of this is from a generally Leftist perspective, without leaning into any given tendency, as I believe the most critical battles now are building up a sizable leftist coalition. Everyone should focus on organizing, unionizing, reading, learning, sympathizing, empathizing, and improving themselves and those around them.

I've never heard anyone argue against personal property. Usually the difference is that Anarchists want to skip the workers' state, while other Communists think it's a necessity to achieve Communism.