Much is made about the title of 'Project Manager' and the need (or not) for that position in agile software projects. I've always felt that the role of Project Manager is different than the role of a Scrum Master even though the roles are often attempted to be filled by the same person. While there may be some overlap in the roles, the Scrum Master is generally focused internally toward the team and the Project Manager is often focused on activities that affect the team but are external to the team. A Scrum Master would be focused on guiding the team through the Scrum framework, helping the team improve their effectiveness, and removing impediments for the team. A Project Manager is often tasked with ensuring that regardless of what is transpiring within the team, the team is aligned with current corporate protocols and constraints. Ideally, of course, the entire organization is using an agile mindset and the role of Project Manager is subsumed by that of Scrum Master. In reality, at least as a transitional move, the two roles are still required during the early stages of an agile transformation.
The role of Product Owner and/or Manager on a sofware project should have the responsibility for making all the business decisions related to scope, budget and schedule because they are supposed to have the most understanding of the value created and the business priorities for the marketplace. Other members of the team (and indeed of the organization) may have different opinions but it is the role of the Product Management discipline to make product business decisions.
In Oil and Gas Facilities construction, the Project Manager is the arbiter of ALL of those same business decisions. It is from the roots of construction/manufacturing that software development took its first organizational steps and so it is not surprising that the title of Project Manager has been confused.