A project manager is focusing on what we have to do. On the outputs like what, for how long, what resources, etc.
The product manager is focusing on outcomes like what clients should get. What should our users be able to do? What level of quality should we have?
Project managers usually screw up by not doing the promised number of tasks by the promised date. Product managers typically screw up by not delivering agreed outcomes.
Now you know why I prefer product focus.