You have to clarify what you mean by "clinical professor". Do you mean a professor who is in clinical medicine as opposed to basic medical sciences, or are you referring to the titles "Clinical Assistant Professor of...,Clinical Associate Professor of...., Clinical professor of ....."?
The "Clinical Professor" title series essentially means they are not really part of the medical school system in what you are used to seeing. They have some sort of agreement or adjunct status with the medical school. For example, I remember that there was this private practice in the Maryland/DC area who had an agreement with Johns Hopkins that you could get "Clinical assistant professor" title from Hopkins and really didn't involve much other than letting medical students rotate through your practice. These "Clinical professor" titles are a dime a dozen and only mean that you are not a traditional faculty, but every now and then you help out in the missions of the medical school in one form or the other. Most people in the know are aware that these are semi-bogus university titles, but it works if you want to show off to lay people.
On the other hand, the real faculty starting with Assistant Professor, then Associate Professor, and finally Professor are standard traditional faculty appointments. Some top medical schools (like Harvard) don't even let people start at the Asisstant Professor leveland they have four levels, the starting level being "Instructor in ...". Then after a few years of clinical work, teaching, research, and publications you may get promoted to "Assistant Professor", in much the same way that at most other medical schools you would have already made "Associate Professor" by the same criteria. It is not unheard of that when people move to Harvard faculty, they get "demoted" by one level for example from full Professor to Associate professor.
There is also another level of complexity added in the last decade or two in many medical schools. The traditional (real) faculty in clinical specialties can be hired in three or four different "tracks". Different medical schools name them differently, but they are pretty much the same thing with various terminology. One such track is the clinician-educator track (the vast majority of faculty are in some incarnation of this track). It involves clinical work, teaching and some research. There are also research tracks. tenure tracks at the assistant professor level have become exceedingly rare these days. There is also a clinical track which minimizes research requirements but is difficult to get promoted. All these different tracks have different promotion requirements and criteria.