What's the difference between research mentor and principal investigator?

In your case, the professor is probably both. PI is a technical term that applies to the person who is given the research grant from an agency or is officially in charge of a research project if internal. There can be one PI or several and they do not even have to be at the same institution. A research mentor is a personal term that applies to the person who you feel is the one that mentored you while learning to do research. It depends on the situation.

For example, in a very large group, the PI may be a professor that never enters the lab or never works directly with the staff in the lab. He only writes and reports on grants. They may have several other professors or post-docs that work for them that actually work with students. These later people may be mentors. Who your mentor is can only be decided by you.