The Maya did not die out, there are several million of them alive today. The Lowland Collapse is not an apocalyptic collapse that most people think. It was a socio-political and economic change. The cities along the coast actually thrived with coastal trade from Veracruz to Honduras.

The Aztec did not take their land, however they have several colonies in the area such as in Chiapas.

The Aztec did not come from the Maya region. They came from a mixture of Chichimec tribes who migrated from the north and native groups already established around Lake Texcoco.

They believed in similar themed gods, but not the exact same gods. For example they both had rain gods with "goggles" and a wind god that was a feathered serpent.