What were the relative strengths of the Copernican and Ptolemaic systems from the viewpoint of the 1550s?
The geocentric model of the universe propounded by Ptolemy was actually quite complex. It assumed that the moving stars moved around the earth along a circular pathway defined by a celestial sphere. In addition, though, as the sphere turned, the planet was actually attached to a smaller sphere fixed to the larger one giving smaller circular motions called epicycles which did an excellent mathematical job of approximating the movements of the planets (including retrograde ones) as seen from earth. A geocentric model is useful for celestial navigation.
The Copernican model of the planets revolving around the sun in circular orbits has the merit of simplicity, but is not useful for celestial navigation, and is inaccurate in predicting the apparent motion of planets because Copernicus did not realize that the orbits were actually elliptical.