Direction and day-to-day choices
A product manager usually works out which user problems matter, how the product supports the organisation’s goals, and what success would look like. They bring together research, customer feedback, and business needs to shape a roadmap. The exact responsibilities vary between organisations.
A product owner is a specific accountability in Scrum. They are accountable for maximising the value of the product and for making the product backlog clear, ordered, and understood. They may do backlog work themselves or ask others to help, but remain accountable for it.
Titles do not tell the whole story
Imagine a team improving an online checkout. A product manager might investigate why customers leave before paying and set a goal to reduce that problem. A product owner might then order the backlog so the team tackles the most valuable changes first. In a small team, the same person could do both. Some teams use only one of these titles.
Neither role does the work alone. Designers, developers, researchers, and other stakeholders bring information and expertise. When a decision is unclear, ask who owns the product goal and who decides the order of backlog items.