For example, when is the "projected" or "expected" finish date of this project? Is there any difference? If not, which one is more common?
The "projected" date implies that some level of analysis or study has been performed to identify that date.
The "expected" date does not imply any such analysis; expectations may be set by any means, or none at all.
To understand the difference, consider the following:
Given the amount of work and the number of workers, our projection is to be complete by January of next year. However, the sales manager has already told the customer it will be done by July of this year, so that is their expectation.