I am an astrologer, and I observed the types of astrology Trelawney and Firenze teach (and presumably practice) are two different types.
Trelawney’s astrology is natal or genethiliacal astrology. It consists of personal birthcharts and what will happen when Mars and Saturn make an angle. As Parvati said, it has to do with individuals and their lives (OP27). It’s 95% of the popular astrology today, and 99% of the books you see today contain something on this subject, as that’s what sells books.
Firenze (and the rest of the centaurs) practice Mundane, or political astrology. This is much older and concerns countries, regions, and the world in general. This is the astrology that Nostradamus used in his predictions. It has little to do with individual personalities, except when they rule (govern) countries or affect the world (like You-Know-Who, and maybe Harry). But the statements about the world is between conflicts and Voldemort will return and the rest and even the famous “Mars is bright tonight” are all attributable to Mundane astrology. Firenze’s burning of mallow root and the rest is supporting mundane predictions with various forms of scrying. It’s similar to what the Ancient Greeks and Romans did. (And possibly the Chaldeans—the original astrologers—but we don’t know that much about their exact practices.)
Anyhow, comparing what Firenze and Trelawney do with astrology is like comparing apples and tomatoes. They speak different astrological languages.