Both systems have the Sun at the centre, but Copernicus stuck to the ancient model of circles and epicycles to explain the planets' orbits.
Kepler on the other hand used new measurements by Tycho Brahe to suggest, after a lot of detailed calculations, that the planets move in elliptical orbits.
The difference between an ellipse and a circular orbit with an epicycle is extremely small in the case of planetary orbits which have a low eccentricity factor. It was not until Tycho came along that measurements of sufficient accuracy were available for Kepler to make his discovery.
Later the elliptical orbits were explained theoretically, after the discovery of the law of gravity and the laws of motion. This eventually led to Kepler's theory being generally accepted as right, which it still is.
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Nicolaus Copernicus proposed a heliocentric model of the solar system, which deviated from the geocentric model in the Ptolemaic system. In Copernicus' model, the Sun, not Earth, was at the center of the universe, with the planets, including Earth, revolving around it.
Ptolemy and Copernicus' ideas about the universe are different from each other in the sense that Ptolemy thought that every celestial object as well as the sun and the moon orbited the Earth whereas Copernicus had the thought that all planets orbited the Sun, while the Moon orbited the Earth.
Copernicus
The difference between Ptolemy's and Copernicus's model was that, Ptolemy's model had the Earth in the middle of the Solar System, with all the other planets (including the Sun and the moons) revolving around it. In Copernicus's model, he had the Sun in the center of the Solar System.
Copernicus