Yes, they mean the same thing. But a couple of nuances:
1. 'get the project started' is slightly more informal.
2. I need to start the project = I, personally must do the project.
'I need to get the project started' could mean you will ask someone else to start the project - it has a connotation of 'I will set something in motion, but I won't necessarily do it myself.' However, it is still possible that it is you who will start the project if you say this.