What eventually caused the Ptolemaic view of the solar system to be discarded was that its math became too difficult. The math for the Copernican view was much simpler. I remember from an astronomy class in college being told that everything, even the equatorial bulge, could be explained in terms of an unmoving Earth with the entire universe rotating around it, but the math is very difficult. On the other hand, it all depends on your context. If you're charting a path from the Earth to Mars you'd do a lot better considering the Sun unmoving and the planets in orbit around it, but if you're charting a path from New York City to Kalamazoo, Michigan you're better off with a stationary Earth.
When you come down to it, Copernicus was just as wrong as Ptolemy, but one step further out. The Sun is not stationary, but is in orbit around the center of the galaxy and the galaxy is flying around through space at the mercy of any other large agglomeration of gravitational bodies. You choose your math to fit the problem.
