Product scope refers to the outcome; the specifics of the product or service that the project is expected to deliver.
Project scope refers everything that has to happen to get the product created and delivered. It also refers to the things that can't happen, to get the product delivered (i.e. scope "exclusions")
The differentiation in product and project scope becomes interesting when you talk about Cost of Quality, for example, because you might tighten down project scope so much (eg to save money, or get it built and deployed into service quicker) that you actually impact product scope to the point that it becomes problematic once deployed (consider the recent flurry of Toyota recalls). The responsible project manager will consider the trade-offs on project and product choices and use tools like the cost/benefit analysis to make solid full life-cycle decisions.
So, a project manager should accept responsibility for both project and product scope.