Copernicus used the same circular-orbit-plus-epicycles system as Ptolemy, just the orbits were centered around the sun (kind of---each planet had its own circle, with the sun only approximately in the middle). The system actually had more epicycles than Ptolemy's and was less accurate. It wasn't an advance in any meaningful sense.

The real breakthrough was Kepler, who dropped the idea that planets moved in circles. It was indeed partly a mathematical breakthrough and the reason Kepler's work took a while to catch on is that people couldn't understand his math at first. But it was also empirical, as Kepler had access to new and much more precise observational data collected by his mentor Tycho Brahe.