I think an important point that hasn't been made yet here is that getting a doctorate is a necessary, but not sufficient, condition for becoming a professor. It depends on the field, but typically when you graduate with your PhD, you don't immediately get hired for a faculty-level position (i.e. professor). So, as a PhD-bearer, you have the right to be to referred to by the title of "Dr.", but you haven't yet been deemed awesome enough to get a faculty job. This, at least, is how it works in North America. Instead, you're (hopefully) hired into a position as a researcher to continue to do work in your field. Many of these post-doctoral researchers are referred to as, appropriately enough, post-doctoral researchers or "postdocs" for sure. Postdocs are "Drs." but not "professors." In my field, it is necessary to go through one or two post-doctoral positions in which your only job is to do research/science (i.e. you have little to no administrative or teaching responsibilities). You just publish like crazy and try to make a name for yourself in your field in the hopes that you will get hired in some tenure-track position eventually, e.g. assistant professor. However, this is far from guaranteed.
Disclaimer: I'm just a lowly PhD student and can only go by what others have told me about the academic career path post-PhD. I'm not speaking from personal experience.
