If you choose to manage a project, it's pretty safe. As the manager, you report. You report on what's happening, you chronicle the results, you are the middleman.
If you choose to run a project, on the other hand, you're on the hook. It's an active engagement, bending the status quo to your will, ensuring that you ship.
Running a project requires a level of commitment that's absent from someone who is managing one. Who would you rather hire, a manager or a runner?