What is an astrologer?
Following the etymology of the word, "astrologer" literally means "one who studies the stars." More specifically, an astrologer studies the stars and their influence on human affairs. An astrologer examines the alignment of the heavenly bodies and makes predictions from it. While an astrologer is concerned with the big picture, his/her understanding of the stars and their influence did not have to be general; astrologer could and did cast individual horoscopes based on the circumstances of the individual (their date of birth, for example).
The reasoning behind astrology is predicated on the idea of a microcosm/macrocosm. The human body was perceived as a system in and of itself. Mapping the solar system over the figure of the human body, it was believed that the head was positioned in the Empyreum, the place of God, suggesting that the head can contemplate the nature of God. At the same time, the bowels, the dirtiest part of the human body was mapped over the center of the earth, believed to be the location of Hell (in the geocentric model). Since astrologers (who were not distinct from astronomers until the seventeenth century) saw this correspondence, the actions of the heavens should also correspond with the actions of the human body.
An astronomer, on the other hand, does not infuse the heavens with this connection. He/she is more concerned with the structure and nature of the heavens; an astronomer does not preoccupy him/herself with the question of how the heavens relate to humans and/or human activity.
I would also simplify their names to "stargazers" or "star readers" for their capacity to predict patterns in the movement of the cosmos and stars in general, and by trying to correlate such pattern to current events.
In other words, an astrologer studies the intrinsic relationship between solar and stellar patterns and their effect on our own planet's patterns of movement to create a relationship between stars, movements, to behaviors, and happenings.