The primary difference is that "astrologer" is the preferred and widely accepted term for someone who practices astrology, while "astrologist" is a less common and generally discouraged alternative.
While "astrologist" isn't grammatically incorrect and has been used historically, it's simply not the standard term used by professionals or enthusiasts in the field today. Using "astrologer" will always be more accurate and better received. Therefore, when referring to someone who practices astrology, always choose the term "astrologer."