A project manager is responsible for the succesful execution of a project. Success is usually defined as delivering the project on time, within budget, to the predefined requirements.
A product owner is also responsible for the succesful execution of a project. He manages a team, determines the requirements, the order in which the team delivers these requirements, and he determines when and how often these requirements are released.
The difference between a product owner and a project manager are the instruments used to manage a project.
Scrum is based on the fact that projects have large number of unknown aspects. Requirements may be unknown, productivity is unknown upfront, technical challenges will be unknown. Because of all these unpredicatable aspects, it’s impossible to draw up a plan upfront which predicts time, budget and delivered features.
A product owner manages the project by focussing on the requirements. What do the customers need? He makes sure that the most important aspects as done first. If you don’t know exactly what you need to ship to your users, and you don’t know exactly how long it will take to create it, you need prioritize.
“Do we still need a project manager if we’re doing scrum?”
Yes, scrum needs a project manager. It’s the product owner.