Within the tech sector, project management refers to the set of competencies and tasks around coordinating projects - a lot of stuff around task tracking, reporting progress, unblocking problems that come up, and so on. Product management refers to a set of competencies and tasks for making sure you're building the right thing to solve the right problem. Tasks like conducting market research, user research, and crunching product metrics/analytics generally fall more on the "product" side than "project" side.

The actual boundaries tend to be a bit fuzzy because a lot "product management" comes down to doing whatever the team needs in order to make sure a good product is delivered, and by necessity product managers often do a substantial amount of project managing.

Typically you'll see people conflating "program management" and "product management" in tech. You don't usually see "project management" in that mix since that tends to refer to a more industry agnostic discipline.

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.