A constellation belongs to the Zodiac if the Sun passes through its place in the sky during the course of the year. There are commonly believed to be twelve constellations, but the Sun actually passes through thirteen. The ancient Babylonians decided to only count twelve because that number was more convenient, and they assigned each constellation to a month of the year, resulting in the twelve-symbol astrological Zodiac calendar. While it might seem a bit strange that they would leave out a whole constellation, keep in mind that there is a possibly valid justification, which is that the Sun does not pass straight through each Zodiac sign, but instead it passes just through the edge of some of them, while travelling directly through the middle of others. Perhaps this missing constellation, called the Snake Annoyer, was only barely in the path of the Sun, so the Babylonians chose it to leave out so there could be twelve.
Besides the fact that there are an unclear number of Zodiac constellations, there is also another factor that makes Astrology a bit uncertain. This is the fact that the Earth's axis has moved since ancient times, and now the Sun passes through the constellations at different times of year than it used to. This means that there is now about a month of discrepancy between the historical times of the Zodiac signs, and the current times, which is confusing, because now, which one is correct? And it will only get worse in the future, because the Earth will continue to point in a different direction over time, which will make the Zodiac fall even more out of line.
Astrology is different from Astronomy because Astrology is concerned with trying to predict the future from the movements of the planets and the stars, and it is only concerned with the Moon, the Sun, five of the planets, and only the stars that are visible unaided from Earth and lie in the Zodiac. On the other hand, Astronomy is concerned with trying to understand why everything happens in the Universe, and although its domain used to coincide largely with Astrology, it now has expanded to stars that are not visible from Earth with the unaided eye, as well as other objects like asteroid, black holes, and nebulas. An astrologer would describe its job as making predictions and studying the motion of planets through the Zodiac, while also comparing these things to personalities or events. An astronomer would describe its job as looking at a distant galaxy and trying to determine what elements are present, or trying to calculate the distance to another star system, or trying to deduce the mass of a black hole by seeing how it interacts with the stars and dust clouds nearby.
Astrology has one thing in common with the scientific method, which is that it makes predictions. However, it breaks from the scientific method in that it does not test them, it simply assumes they will all be correct. A scientist would never assume they are right until they have shown that their prediction hold up with what it predicted.
If the thirteenth Zodiac sign were reintroduced into Astrology, it would completely throw off Astrologers because there would have to make an entirely new list of combinations of celestial bodies that includes this constellation, as well as redesign the Astrological calendar in order to accommodate a thirteenth month in the year. Astronomers, however, would not be nearly as shaken, as they already know where all the stars are, and no stars or groups of stars are of any special importance to them. They will continue making observations and taking measurements and readings, while the Astrological community is being overturned by the introduction of a thirteenth member of the Zodiacal calendar.
Well those are my two cents on the subject. More like two dollars, I might also add.