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...