I used autospinstaller to install Project Server 2013 also. The Project Server Service Application was installed and the Project Web App (PWA) Instance was provisioned (along with the database). However, when I access the PWA instance it doesn't show up
as a Project Server site and the PWA Settings menu item doesn't show up in the Site Actions menu. I ran the Set-SPProjectPermissionMode cmdlet to see if that would help as suggested by this link:
but that didn't work.
However, I created another PWA Instance using the same Project Server Service Application and ran the Set-SPProjectPermissionMode cmdlet and it works as a PWA instance.
Can you indicate where the issue could be with the PWA instance provisioned by the autospinstaller?