One thing, perhaps the only thing that anarchists and socialists agree on is the necessity of working toward a society without capitalism and without the state. Both are seen as antithetical to an egalitarian society in which human beings can achieve maximum personal freedom.
The path(s) to this end differ between the two groups. Socialists see capitalism as the key social structure, insisting that in the absence of capitalism, the state will wither away.
Anarchists hold that it is the state that allows and supports capitalism, and that if the state is removed, capitalism cannot stand.
Both of these views jump from the present to a future society devoid of the state and capitalism and ignore the process of removing the state or doing away with capitalism. A wave of the social magic wand, poof, they're gone and the anarchist, or socialist society is secured.
Unfortunately for us in the present, living with the state and capitalism in full hegemonic control, we must work to achieve the society we envision. We must build the new society from scratch, by destroying the existing state and economic system, or by building the alternative alongside the dying remnants of the existing society.
In reality, inconvenient as it is, we cannot "destroy" the state any more than we can "get rid of" capitalism. These are social structures fully embedded in Western society that cannot be gotten rid of without destroying society as a whole, reducing all to chaos.
Furthermore, we cannot create an egalitarian society using oppressive measures, such as violent revolution. We cannot use the tools of the state to replace the state. If we violently take over the state, we become the violent state. Once initiated, a violent takeover cannot be set aside for a benign, nonviolent social structure.
Once the vacuum of power is filled, it cannot be emptied. We must do away with paths to power, not fill them with our own, however enlightened rulers.
Anarchy is no rulers. Not even our own rulers.
Weeks' Antiques will deal with actions we can take toward the new society. Here I will deal with the various theories of getting from here to there. Hayduke Ponders will deal with the philosophical/spiritual underlayment of action and theory.