The thing is, there is more than one Geocentric system, there's the Ptolemaic system, with the sun and planets revolving around the Earth and then there's the Tychonian system (named after the famous astronomer Tycho Brahe, who invented it in the mid 16th century), with the Sun and stars going around the Earth and the planets going around the Sun, it looked like this (blue orbits go around the earth, orange ones around the sun):
While it may seem hilariously stupid to us now, it was actually more scientifically valid for a century.
In terms of accuracy, it's literally identical to the Copernican system in terms of what you see from the Earth. And until Newton published his theory of gravity (a century after Tycho published his), neither system had an explanation for why the celestial bodies move the way they do, except for "God willed it".
However, the Copernican system had one glaring flaw, that Tycho noticed:
If the Earth moves around the sun, then the angles of the stars at opposite ends of it's orbit should be different (this phenomena is called stellar parallax). We now know that stellar parallax exists, but instruments precise enough to measure it didn't exist until the 19th century. However, even the smallest visible star has some apparent width. We now know that it's an optical illusion, but the knowledge of optics required to discover that didn't exist until the 19th century (again). Given the calculated orbit of the Earth, Tycho determined that at the smallest possible distance with no visible parallax (for the measurement precision of the day), the dimmest visible star would have to be bigger than the Solar System and the biggest way bigger than that, which he considered preposterous.
So basically, it took Newton, the discovery of Stellar Aberration in the 18th century and the invention of the Foucault pendulum in the 19th (yet again) to prove the Earth rotates, to finally bury the Tychonian system.
I think that the popular perception of the Heliocentrism vs Geocentrism is that of Galileo muttering "And yet it moves" before the stupid Inquisition(Apocryphal btw, there's no contemporary evidence of it happening). But Galilleo was like Columbus(1), right, but for the wrong reasons and has an extremely in-accurate mythos(2) due to what amounts to propaganda(3).
(1) This is an aside, but demonstrates a similar popular misconception. Columbus was basically a fool who got extremely lucky. Contrary to popular opinion (3), he didn't discover the Earth was round against the opposition of church troglodytes who thought it was flat. Eratosthenes had correctly calculated the circumference of the Earth in the 2nd century BC and his writings were widely known and accepted in the church and Europe in general at the time. Knowledgeable people derided the idea of sailing west to reach India, because they knew approximately how far east India was and given how far west you'd have to sail (the width of the Atlantic + America + the Pacific), they knew no ship could possibly make it. Instead, he thought he could do better and incorrectly re-calculated the circumference of the Earth, showing it to be significantly smaller than it actually is and thus the voyage was possible. He used his considerable powers of persuasion to convince Queen Isabella (who was clueless in these matters) to fund his voyage. If the Americas didn't have the good luck to exist, he and his entire crew would have died of thirst and disappeared without a trace. To his dying day, he thought he had discovered India.
(2) Galilleo had the kind of personality that could best be summarized as "contrarian douche-bag". If he thought he was right, anyone who dis-agreed with him was an idiot and he did not hesitate to insult them. He wasn't imprisoned for saying Copernicus was right, he was imprisoned because in his famous book, "Dialogue Concerning the Two Chief World Systems" (which was written in the form of a dialog between philosophers), the opposing view-point is espoused by a foolish character called Simplicio (overtly named after the philosopher Simplicius, but an Italian double-entendre for Simpleton/Moron), who had characteristics identifying him (for everyone who knew him personally) as Pope Urban VIII, Galileo's former friend and patron, who opposed the Copernican system (he also pulled the straw-man trick by opposing the Aristotelian system to the Copernican one and pretending that the Tychonian system did not exist, although it was the most popular theory at the time). So basically, as far as all the important people in Europe were concerned, had publicly called the Pope a moron with the barest fig-leaf of plausible deniability. At the time, the Pope was also functionally a head of state, with lands and an army and that kind of insult would have have been ample grounds for any nobleman from Baron upwards to kill Galileo on the spot, given the mores of the day, but all he got was house arrest in a mansion for the rest of his life. P.S. Galileo actually spent way more ink arguing for his (incorrect) theory of the tides being caused by the Earth going around the Sun.
(3) This all happened during the Reformation and the wars of religion, so Protestant intellectuals jumped on any chance to portray the Catholic church as venal idiots, holding back scientific progress, so any nuance was (very probably deliberately) omitted from their literature.