Tenochtitlan slowly became Mexico City as the Spaniards built European buildings on the foundation of Mexica ones. The classic examples are churches and cathedrals, which were frequently built over temples. Today, the old city is basically under the modern one, and frequent excavations keep uncovering parts of it. The Templo Mayor and La Plaza de Tres Culturas are the best examples. However, small temples, schools, and other buildings are constantly being discovered.

The Spanish fundamentally changed the city and its environment through a process known as the 'Desague,' which was the draining of Lake Texcoco. For the Mexica, Tenochtitlan was an aquatic city, but the Spanish had trouble dealing with the water, and so spend two hundred years and vast amounts of cash trying to get rid of it. One of this acts many consequences is that the environment in and around the city now bears little resemblance to its Precolumbian predecessor. Now some planers and architects want to bring it back, though this seems unlikely.

For more information on this process see:

López, John F., “In the Art of My Profession”: Adrian Boot and Dutch Water Management in Colonial Mexico City’, Journal of Latin American Geography, 11 (2012)

Candiani, Vera S., Dreaming of a Dry Land: Environmental Transformation in Colonial Mexico City, (Stanford: Stanford University Press, 2014)

Moody, Barbara E., The Death of Aztec Tenochtitlan, The Life of Mexico City, (Austin: University of Texas Press, 2015)

The city suffered major siege damage during the conquest, but it survived. The Spanish quickly made it the capital of the region calling it the 'ciudad de Mexico-Tenochtitlan'. Eventually the Tenochtitlan was dropped. We now know it as the Ciudad de Mexico or Mexico City.

The central core (the sacred precinct of the Aztecs) was razed. A church built across from the old templo mayor. Civic buildings rose up over ruined palaces and private lots were parcelled out to conquistadors. Beyond that the city slowly rebuilt. Because of population loss some areas remained abandoned but to the west and north of the center thousands maybe tens of thousands of Aztecs continued to live.

There is a recent book by Barbara Mundy called The Death of Aztec Tenochtitlan, the Life of Mexico City that does a great job telling this story and mapping the changes.