I quite like this summary from https://www.skeptic.ca/Anarchism_Introduction.htm because it points out a few key differences.
The basic tenet of anarchism is that hierarchical authority -- be it state, church, patriarchy or economic elite -- is not only unnecessary, but is inherently detrimental to the maximization of human potential. Anarchists generally believe that human beings are capable of managing their own affairs on the basis of creativity, cooperation, and mutual respect. It is believed that power is inherently corruptive, and that authorities are inevitably more concerned with self-perpetuation and increasing their own power than they are with doing what is best for their constituents. This is equally true in Totalitarian and Communist states as it is in Democracies. Anarchists generally maintain that ethics are a personal matter and by definition must be based on concern for others and the well being of society as a whole, rather than upon laws imposed by a secular or religious authority (including revered laws such as the U.S. Constitution). Not unlike existentialist philosophers such as Jean Paul Sartre, most anarchist philosophies hold that individuals are responsible for their own behavior, life goals and ultimate purposes in life. The anarchist needs no one to tell him what he ought to do and how to create a meaningful life. Paternalistic authorities foster a dehumanized mindset in which people expect elites to make decisions for them and meet their needs, rather than thinking and acting for themselves. When an authority arrogates to itself the right to overrule the most fundamental personal moral decisions, such as what is worth killing or dying for (as in military conscription or abortion), human freedom is immeasurably diminished.
Anarchists acknowledge the connection between various forms of oppression -- including sexism, racism, heterosexism, classism, and national chauvinism -- and recognize the futility of focusing opposition on one form of injustice while others continue to exist. Anarchists believe that the means one uses to transform the world must be in accord with the ends that one hopes to achieve. While anarchists disagree about strategies and tactics, including the need for formal organizations and the use of violent action to overthrow existing oppressive institutions and injustice, most agree that the focus must not be on merely destroying the current order, but on fashioning new, more humane and more rational alternatives to take its place.
While many anarchists value cooperation, communalism and collectivism, anarchists reject both the plutocracies of capitalist states based exclusively on greed and envy and the totalitarianism of the existing and recently fallen communist, or more accurately Marxist-Leninist, states. The rift between anarchists and Marxists developed as early as the 1870s as anarchists perceived that the Marxists were perpetuating authoritarianism under a different name. Marxist-leninists groups have traditionally emphasized the need for a vanguard party and the dictatorship of the proletariat, ideas which are fundamentally opposed to the anarchist focus on anti-authoritarianism and maximum individual freedom. Although orthodox Marxism predicts that the state will "wither away" with time, we have repeatedly seen in communist regimes a consolidation of state power and its attendant repression and insistence on conformity. The same oppressive authoritarianism has emerged in capitalist democracies.
I just re-read your question btw, and realized I kind of answered the wrong question here, because the quotes focus on the differences in the means rather than in the end goal. But it's still good context, because communism has so far never been able to move beyond the authoritarian stage, so you could reasonably argue that the actual end state of communism is an authoritarian state with a socialist economy, especially if they have to co-exist in competition and conflict with capitalist states. So the end goal of the state "withering away" is more or less impossible to achieve via communism while western hegemony exists.
“It is no measure of health to be well-adjusted to a profoundly sick society.” Krishnamurti