The key responsibility (and purpose) of a project manager is to manage and organize the execution of a project so that it achieves the goals which were defined for that project.

The key responsibility (and purpose) of a product manager is to provide a coherent vision of what the product needs to be successful and define what the ongoing projects' goals should be to effectively improve that product.

Engineer - how to build; project manager - how to organize building; product manager - what to build.