If you adopt a geocentric reference frame, then the Sun appears to rotate around the Earth, and all the other planets also appear to follow epicyclic paths around the Earth. You can predict the motions of every planet and star equally well, the equations are just different.
So again, why is one more 'correct' than the other, besides calculational simplicity?
Geocentrism and Heliocentrism are not just calculation frameworks for making observations of the sky. They are also an attempt to create a model of the fundamental rules that govern the heavens, and then to create observable predictions based on those rules.
Heliocentrism is the better model of the two, because the paths of the planets are dominated by force of gravitational attraction to the sun. This force is what you model, regardless of your frame of reference. You can use this model to create observational predictions for any point in the solar system, but the forces that your modeling are essentially the heliocentric ones.
But to your larger point: In my grade-school science class, we were all taught that heliocentrism was itself supplanted by the discovery that the sun is rotating around the galaxy, and that modern cosmology doesn't center around anything at all. Heliocentrism is still more accurate than the geocentrism that it replaces, but is still regarded as a stepping-stone to modern theories.
The simple answer of why one is more correct, is because it's deemed as correct by the scientific community, which comprises the scientific experts on that issue.
The larger issue you raise is one that was especially prominent in mid-20th century philosophy of science, specifically in Thomas Kuhn's concept of a "paradigm shift". A radically new scientific theory does not displace an existing theory because it is more "correct", since ad hoc modifications (e.g., epicycles) can always be made to the existing theory to account for new evidence that appears not to fit. However, at some point the existing theory will come to have so much baggage, and the new theory will explain the world so much more elegantly (e.g., with calculational simplicity) that the new theory will overthrow the old one and become the new "paradigm". Note that this is not because the old theory could not accommodate all the facts; any old theory can if you modify it enough, but at some point when you have to make all sorts of strange adjustments to the existing (old) theory it just becomes wildly implausible when compared with the new theory. At some point scientific consensus changes and the the new "paradigm" is accepted. You can read more about this at Wikipedia:
I've always thought of the difference between geocentric and heliocentric to be an ego thing. Setting Earth as the center of the Universe works great, for everyone on Earth. It enforces the idea that Earth, and through that, Humans, are so important that the Universe is really centered on the Earth.
Maybe this is the case, maybe its not. To assume that the Earth is the center because of us is pretty egocentric.
Heliocentric places it in a more objective way, I guess? Changes the reference point to be consistent with the known laws/theory/whatever of gravity.
It could also be a scaling thing as well. Applying the equations to planets while setting a reference point based on our sun would provide equations that are more easily adaptable to other galaxies or solar systems that also have planets rotating around the sun. Sure, we could try and create formulas that describe the motion of other galaxies in reference to the Earth, but that would fall apart if the Earth stopped existing.
I'll admit I'm way out of my depth here! But wouldn't the fact the planets experience orbital decay that ultimately cause them to be pulled into the star they are orbiting, mean that heliocentric is "more correct" when discussing a planet's relation to a star?
You could still calculate that with an earth-centric approach, it just might not be fun. As far as the physics go, it doesn't care which point or body you pick to be non-moving - the math all works out, it's just relative to a different point. And with that, there's no spot in the universe you can point at and say "That point is not moving", so there's no "fundamental zero" we can compare everything to.
Perhaps the point is that "more correct" is subjective. Both approaches will give you the right answer, it's just that heliocentric makes finding the answer easier.
