To elaborate Martin's answer slightly, the Copernican (heliocentric) model is indistinguishable from the Ptolemaic (geocentric) model in its final form.
This is because both models in their final form are actually mechanically equivalent.
What distinguishes planetariums/orreries based on each model is that the geocentric planetarium attempts to explicitly describe the movement of the heavens as seen from Earth (so that things like retrograde motion are explicitly seen in a geocentric planetarium).
Whereas in the heliocentric planetarium, the movement of the Earth together with the movement of other elements makes it more difficult to infer the observational reality we see from Earth.
The heliocentric planetarium effectively demonstrates things as would be seen by a person standing on the surface of the Sun - which is not an observation that has ever been made in practice.
Nowadays, the heliocentric model is regarded as the "simpler" one, but in fact the central importance of Earth to humanity, and the fact that historical interest in astronomy was primarily driven by a desire to model what was seen in the sky (not to represent some abstract cosmological view), meant that for a long time the geocentric model was regarded as the simpler and more obvious description.
It's also important to note that both models actually date from antiquity - the heliocentric model doesn't actually originate with Corpernicus, but is at least as old as Ptolemy who was also familiar with it.
What probably harmed the geocentric model most was that improvements in telescopy meant that irregular orbits and patterns of eclipses were more readily seen and precisely quantified, which increased the mechanical complexity of geocentric planetariums that accounted for them. Better telescopes also meant that an increasing number of elements had to be accounted for.
The "equants" and "epicycles" and so on of the geocentric model, which exist to represent very specific motion which is apparent/relative to Earth, but are unreal in the sense that they do not represent a general description of astronomical movements, massively increases mechanical complexity, and as more elements are added there is a risk that the sweeps of moving linkages and supports start to conflict with each other.
It was also not entirely clear in the past whether such apparent movements in the heavens were "real" or not, but the increasing complexity and arbitrariness of the geocentric model over time, led many to doubt. In that sense, there is a certain philosophical side to the geocentric model which has been falsified - the question of whether it describes something "real", or whether it only describes what we see.
Eventually designing such machinery to represent the geocentric model became too much, so that the geocentric approach began to fail in its own terms (of providing a complete description of the observational reality from Earth, in the form of some conceivable machine that demonstrated the model in motion).
By contrast, a planetarium that implements the heliocentric model is mechanically very much simpler, and the movements of the planets relative to the Sun are seen to be very much simpler and more regular, even though relating it to any earthly observation involves additional reasoning.
