What are the roles and responsibilities of a project design engineer ? how is it different from the roles and responsibilities of a project manager?

Let's assume that the project has been given the green light and that the management structure is in place at executive level. There  is the Project Sponsor, and Director at the Board level.

Oncue the go ahead has been given an experienced Project Manager is appointed. He then assumes control of the project from a management standpoint. He then has to assemble his team based on the objective of the project. The team led by the project manager will scope the project, plan and then launch. Monitoring and controlling the project will also be part of the team's responsibility. At the end the project will be closed by the team within agreed parameters.

Clearly the role of the Project Manager is to manage the project utilizing the team. The PM will manage people and effectively deal with issues that will arise. He has overal responsibility of the project and is accountable to his higher management.

The Project Design Engineer's role will evolve as the team will devolve responsibility of technical specification design to the Design Engineer. Subsequently it will be the role of the Design Engineer to be supervising the execution of the various stages because of his intimate knowledge of the requirements as he would have had the primary role of scoping the project. The Design Engineer should then report to the Project Manager through the team. 

As far as my experience goes in managing large projects this how I see the PM's role and the Project Design Engineers ro and this how I managed projects. Let me add that the PMO has to be fully resourced to meet the needs of the Project.

The PM manages people and higher level issues, administration, etc whilst the Design Engineer ensure the project is done to specifications. Both are very important to delivery.