Project Manager is a manager, it's a role from a formal process (waterfall). Product Owner is not a manager, it's an agile role.
You can't compare them directly.
P.S. In real life PM/PO are almost the same -- as people usually don't care about following metodology. PM/PO is just a kind of team's boss.
I support 4 different software solutions each part of a different product team and each having their own product manager. PM is definitely not my boss, just an unnecessary middle man making the total cost of tech more expensive.
P.S. In real life PM/PO are almost the same -- as people usually don't care about following metodology. PM/PO is just a kind of team's boss.