The difference between project management and product management is like comparing a member of an concert band and a jazz musician. Both have to be masters of their instruments, but while the concert musician is measured by how well they play the written music, a jazz musician is evaluated on their ability to improvise. They have to not only understand the written music, but they have to understand how the music is constructed. They have to listen to what the other players are doing and adapt and improvise to be successful. With jazz, people pay to hear the players, not the charts.
Project managers build a plan and then execute the plan. Product managers build a plan, but then have to have the skills and willingness to adapt as they learn. The outcome is the point, not the plan. What worked with one product team won’t necessarily work on the next … and that’s what makes the job fun.