The biggest objection to Ptolemy's model in Ptolemy's time would have been the fact that the model does not make use of Aristotelian uniform circles, nor in Ptolemy's model is the Earth placed in the exact center of the universe.
As University of Texas scholar Richard Fitzpatrick points out, geometrist and philosopher Aristotle argued that the "heavenly bodies should move in single uniform circles" as the circle is the most divine geometric form (Fitzpatrick, "Ptolemy's Model"). Aristotle also argued that the Earth was in the exact center of the universe. However, Ptolemy noticed anomalies in the motions of the planets that did not quite fit Aristotle's philosophical views. So, in order to still make use of the divine circle while also explaining the anomalies, Ptolemy very creatively designed a model that made use of circles, even though they were not uniform circles, and made use of placing the Earth in the center, even though it was not dead center.
Ptolemy's model made use of two different circles. The first is called the deferent. All the planets move around the earth in one large circle called the deferent. However, one large circle does not completely explain planetary motion; therefore, Ptolemy also designed his model using an epicycle. The epicycle is a smaller circle that each planet is moving in as it also moves in the larger circle around the earth. It helps to picture smaller wheels on a circular mechanism, with one point on each wheel representing the planet, spinning as the larger circle rotates. Therefore, since Ptolemy's model makes use of two sized circles both doing different things, we can easily see how his model defies Aristotle's concept of uniform circular motion.
In addition, Ptolemy's model also makes use of the equant. Since, Ptolemy accounts for the anomaly of the non-uniform motion of the planets by placing the Earth off-center, he re-establishes uniform circular motion by the use of the equant. The equant is the center of motion, which is separate from the center of the deferent. The planet travels around the equant in uniform circular motion, but since it is actually off-center, the motion actually creates the ellipses that we use in our model today ("Ptolemy," ThinkQuest, Educational Foundation).
Hence, Ptolemy's model would have been criticized in his day for not actually using Aristotelian uniform circular motion and for also not placing the Earth in the actual center of the universe.