They are more than "slightly" different. A product manger is a creative professional. A project manager is an administrative professional.
If your company is using the sprint model, the Product manager is not doing much of project management.
At the end of the day, what a PM does depends on the company and how they use PMs
In practice I'm not sure how different they actually are. Typically when someone is desperate to say how much better/harder their title difference is it's because it's not really obvious otherwise.
The project managers are in the engineering department and interface with the engineers, suppliers, operations, etc. Most have engineering degrees.
They work out most of the project documentation together, such as the specifications, budget, schedule, etc.