What is the difference between "research fellow" and "research professor"?

These terms have different meanings in different countries. You have not specified a country or countries.

In my area of the western U.S., a fellow was someone, man or woman, who got financial help while continuing to climb the steps of academia. They were generally what we call residents in medicine. Past internship, often for years/ That is not a negative, except that hours can be long.

A professor is a step above or quite above, in seniority. In my memory, most of those folks had passed lengthy board exams, and been published, usually more than once, depending on whether or not they were first author and so on.

Don't trust me, things may be different elsewhere or different from when I knew it.

Adds that I worked at a university in California. I have no idea how matters develop or are described elsewhere.