The product manager and project manager roles are quite different from each other.
Products and Projects
A "product" is a good or service that is offered to customers. A "project" is a set of tasks and activities that need to be planned. This set of tasks must have a set duration, start date, end date and a related cost or budget.
Straightaway, it should be easy to see that role of the product manager is more concerned with getting the product right. The product manager makes sure that the research, vision and experience of the product is something that customers will value and buy, and also that the product has a valid business model.
However, the role of the project manager is more concerned with making sure the that the project is sufficiently staffed and that the necessary personnel with the required skillsets have a robust, detailed plan to complete the project on time and within the budget. This plan will be monitored to make sure that any faltering in the completion of tasks will be noticed and addressed.
No Project Manager
In smaller companies, you may find that there is no project manager. In these cases, the product manager will be the designated manager who keeps an eye on the team's progress for the given task. Note that sometimes it doesn't make sense to have a project manager in a certain team. Particularly, in the early stages of designing a product or investigating a product idea, the team may not know yet what product needs to be built. Managing the building of a product so that it releases on time and within the budget doesn't make sense if the team has no idea what the product is yet. In this scenario, the team is focused primarily on learning and experimentation.
Agile teams (e.g. teams using SCRUM or Kanban) also typically do not have a project management function in the team. There may be project managers higher up in the company's organization. If the tech department consists of lots of individual Agile teams, the coordination of these teams may require project management as part of the release process.
With large, well-established products, project managers may work in tandem with product managers. It is also true that on very large tech teams, it is common to have separate people focusing on product and project management. The relationship between product managers and project managers is often very good; both are glad they don't have to do all of the work themselves!
In the chart below, there is a comparison of the product manager and project manager's responsibilities. As you can see they are quite different.
The product manager is responsible for conducting product research, creating a product vision, validating the business model and any assumptions, and working with the team to deliver value to the customer. The product manager sets targets, goals, and desired metrics, and ensures that the roadmap of the product's planned features delivers on the product's vision and promise.
The project manager is less concerned with whether the proposed product vision is the best one available or whether the company is following the best strategy. Rather, they are focused on the project - the tasks and activities required to build features and deliver the roadmap.
The project manager is responsible for putting together a plan that sets the project's scope and identifies the resources, cost and timelines/delivery date for the project. The project manager ensures that the correct resources are available and that any risks or dependencies are carefully managed. They also track and monitor the delivery to be able to step in if delivery is late.