What is the difference between capitalism and the free market?

I don't think you can escape circular logic because the terms are vague and overlapping. The focus on private owners is also a dehabilitating diversion. The core is economic competition. Any self-interested actor will try disable competition through various means, including corruption of the public sector, so public/private loses meaning quickly in practice. A focus on private also means that the entirely rational strategies that China takes of locking in natural resources on a global scale and building long term competitive infrastructure are not open to the less competitive economic strategies championed by the US.