I thought the main difference between anarchism and libertarianism was that the libertarians were sell-outs who were cool with statist government.
Private property is a different matter altogether. There are plenty of anarchists who believe in strong private property rights. Anarchism is a very large tent. The one thread that binds them is a desire for non-physically-coercive legal structures for nonviolent behavior**. In practical terms that means no prison for civil violations of the local law. Everything else is basically up for grabs, including the meaning of "law".
** I would think, actually, that preservation of a private property right naturally follows from almost any anarchist school of thought.