The properties of the cation vacancy and the Te antisite, two dominant defects in CdTe and Cd1-xZnxTe alloys grown in Te-rich conditions, are examined using first-principles calculations. First, the structure, electronic levels, and migration paths of V-Cd and Te-Cd in CdTe are studied in detail. Additionally, we analyze the evolution of the stability and electronic properties in Cd1-xZnxTe alloys, taking into account both the role of alloying in the position of the ionization levels and its effects on the equilibrium concentration of those two defects. It is shown that the formation of cation vacancies becomes progressively more favorable as x increases, whereas Te antisites become less stable, backing the trend towards p-type conductivity in dilute Cd1-xZnxTe.