You will get many academic and theoretical answers to this question, but the real answer, I believe, is that all organizations blend a little project management and a little traditional management in their organization. Often “traditional” management is called “operations management” and was developed around manufacturing business models.
A classic operational manager will try to repeat the same basic process, like manufacturing a product, over and over again. They try to optimize for cost, quality, and other variables. I have worked in companies like that. Often they have clear, separate departments, like Sales, Marketing, Operations, Manufacturing, HR, and so on. Each department interacts in a limited way, and typically each tries to optimize their group towards a certain goal. Sales and Marketing optimize for revenue. Ops for production capacity and efficiency.
In a projectized or project management organization, the main unit of work is the project. Projects have a clear beginning and end. Consulting companies often run this way. Each consulting engagement will draw in the needed experts into the team, and when the engagement ends, the team disperses. They try to do the best they can with the project, and are not necessarily looking to repeat that project ever again.
Both of these descriptions are stereotypes, though, not reality. Every manufacturing company has projects. Even repeated production processes can be managed as projects, because each production run has a clear beginning and end. Even projectized organizations need some operations expertise. The consulting companies need to run their facilities, accounting, and other repeated processes, and optimize them for efficiency.
Rather than looking for a PM organization vs. a manufacturing organization, I would recommend looking at each company and seeing how much it takes advantage of each style of management. Some companies really emphasize their project management. Others emphasize operations management. Some try to do both. Almost every company can profit from both styles of management.
Some quality-improvement disciplines, like Kaizen and Six Sigma, can help join the two points of view. All of the quality-improvement methods are used by both project managers and operations managers. The academics from each field both claim that quality management is essential to both camps. Many other topics like leadership and motivation are also shared.
My advice to you is to look at your own situation. Is it more like a project? Clear start and end? Unique results? Different every time?
Is it more like a regular operation? Doing similar work over and over? Repeated processes that must be honed to perfection?
Decide where you want to improve, and where you want to focus. Personally I consider myself a student of both operations management and project management. Most of my professional experience is with projects, but I have used many operations management techniques in many situations.