Product is closer to user experience. Just like the director knows a script inside out, product knows customer problems inside out. Just like a director knows what happens in script writing and editing, product knows what happens in the requirements and testing/QA stages, but isn't entire responsible for it.
This is different at every organization. I have seen companies that make PMs 100% about profit and market, who then delegate everything else down to others. I've seen ones who work exactly as you describe. And I've seen one where Product manager == Jira monkey.
Based on my experience, while there is a lot of variety, most PMs do worry about the markets and profits as part of their job, even though other teams do the actual sales and marketing. Those concerns are part of understanding the product - who is the market, what are they willing to pay, what monetary value aligns with the solution value. Those answers help drive decisions on roadmap prioritization.
A product manager is more like a movie director. She protects the product from feature creep, remains in close contact with users/stakeholders, chooses which bugs to prioritize, etc.
Often there's overlap between the roles but they'll split it up into specialty roles.