The Aztec empire was headed by Moctezuma when the Spanish had arrived to colonize. The Aztecs were also just over a large civil war, so the entire empire was weaker than usual. When the Spanish arrived, they brought diseases that the natives hadn't seen before, like small pox. Moctezuma had tried to treat the Spanish respectfully and graciously, just how you'd treat a god. Cortez, however, wanted gold and more than what the Aztecs were giving him. When they tried to convert the Aztecs to Christianity, Moctezuma had refused and the Spanish attacked. They killed Moctezuma and then continued to colonize and treat the Aztecs badly.
The Inca empire was under Atahualpa, their last emperor, during their fall. The Spanish had come and, like Moctezuma, Atahualpa treated them kindly. Atahualpa was killed as a plan by the Spanish to gain his empire. They then stopped the Inca from practicing their daily lives as usual. They then proceeded to teach them Christianity and turned them into a Spanish colony. The Inca were also new to the diseases the Spanish brought and had were killed vastly by those diseases.