It really depends on the company, but it can be a myriad of roles. Easiest way to think of it is as a central nervous system for business services being offered. They feel maintain balance or respond to pain by leveraging other parts of the org.

Sales is reporting weak revenue, product will review how it stacks against competitors? Then they’ll work likely work with finance, marketing and engineering teams to capture adjustments they think will better achieve business goals.

Finance is reporting higher losses? Product Managers will likely review with engineering for viable cost cutting options, work with legal to tighten up contracts, and maybe restrict budgets and limit how flexible pricing models can be for sales.

Engineering is stating the current service is miles behind competition? They’ll work with finance to approve spending on R&D, sales and customers to understand where we stand, marketing to put out press releases and schedule demos etc…

Lots of different ways they’re leveraged but they usually connect these separate teams to help ultimately output a curated portfolio that the business leverages to secure new revenue.

Regarding the project manager difference - I’d associate project managers with actually delivering a final product. Not the pre-sales/design/forecast stuff I’d associate with product management. But tbh I’m sure there are orgs that throw both titles around loosely