The majority of the diet of common people would have been in the form of corn gruel called atolli which could be served in a variety of ways, but usually would have been savory, served with beans, chili, and salt. Nowadays atolli is still eaten in Mexico, but almost always sweet. People across Mesoamerica most certainly ate tortilals and tamales, though tamales were more of a Maya Area thing and tortillas were more of a Central Mexican thing (though both areas seem to have been familiar with both). In fact, one of the great enemies of the Aztecs was Tlaxcallan, which translates to "place of tortillas."

One of the most major differences between the cuisines of Pre-Hispanic Mesoamerica and modern Mexico is the abundance of meat and where it came from. The average Mexican absolutely eats much more meat than the average Pre-Hispanic Mesoamerican (the same was generally true for everyone in the world at the time, relentless meat consumption is very modern), with most of that being pork, beef, and chicken. All of these are Old World domesticates, so obviously they didn't have access to cheese from dairy cows. Mesoamericans had access to domesticated turkeys and dogs, both of which were major cuisine animals (and both were highly prized by Spaniards). The breed of dog in question is not known. It wasn't the Chihuahua, but a much larger breed that seems to have been driven to extinction (or close to it) by Spanish consumption. Aside from domesticates, native people ate a dizzying variety of wild animals such as deer, peccaries, armadillos, iguanas, axolotls (!!), and dozens of species of birds, fish, reptiles, amphibians, and insects. As far as the Aztecs are concerned, Lake Texcoco was a very fertile environment for fishing and foraging aquatic animals, so they had a lot to work with to supplement their maize-heavy diets.

The wealthy would have eaten more meat than common people just like in other parts of the world at the time. Our best source for this, and most features of Aztec society, is Bernardino de Sahagún's Florentine Codex (its modern name). He describes various foods that the wealthy ate, as well as quasi-recipes and lists of ingredients prepared for banquets. Basically, wealthy people would still primarily have eaten a corn-based diet, though in a wider variety. Instead of mainly consuming gruel or one kind of tortilla, they would eat many kinds of tortillas of different sizes, colors, and thickness. Tamales were also served in a similar variety, though they would also have a large variety of fillings.

Sahagun says that most meals and banquets served by the wealthy would consist of tortillas and tamales served with sauce/stew/casserole (made with meat from various animals), then a serving of many different kinds of fruit, then finally, chocolate (served as a drink). These sauces/stews/casseroles (frankly, the lines between them could be blurry) could be made with fowl, turkey, dog, axolotl, fish, locusts, maguey worms, or other animals, or a combination. One quasi-recipe describes a layer of dog meat placed under a layer of turkey. They would generally have beans, chili, tomatoes, and various fruits. These dishes could also be vegetarian if desired, though "vegetarianism" wasn't a thing as far as I can tell. Chocolate was especially seen as a luxury item, and as such it was given an important place as the final "course" of the meal.

Street food, ever popular in Mexico, was a huge thing in the massive markets of Tenochtitlan and Tlatelolco. Sahagun describes a large amount of tortillas served with various sides, from honey, to turkey eggs, to cactus pears. Mainly though, these food stalls served tortillas and some kind of sauce that was generally made with a base of chili and some other fruits or vegetables. Something resembling modern guacamole or salsas roja or verde wouldn't be out of place in these situations (though they wouldn't have included the Eurasian onions that are now ubiquitous in Mexico). Also extremely common as a street food would have been atolli in various varieties.

There's a lot more that can be said about Aztec cuisine. If you're interested in looking at the primary sources directly you could check out Sahagun's text or sections of Bernal Diaz Del Castillo's description of the Conquest (he discusses cuisine on occasion). These aren't exactly accessible texts, however. Instead I highly recommend America's First Cuisines by Sophie D. Coe. In addition to the Aztecs she also discusses the cuisines of the Maya and Inca.

I'm guessing a big innovation would have been the introduction of cheese?

u/Regalecus has already discussed the innovations in ingredients, with cheese and other dairy products being an important one. Plentiful meat, rice, and wheat are also important components of modern Mexican cuisine that are post-Columbian introductions.

We also see new cooking techniques. Traditional cooking techniques included boiling, steaming, grilling, roasting, and dry-drying on a griddle (tortillas fried on a comal, traditionally made from clay).

The two new techniques are baking in an oven and frying in oil. Pre-Columbian cuisine does not appear to have included any cooking oils; the Spanish introduced olive oil (and olive cultivation, aimed at developed a thriving olive industry, but the Spanish government had issues with that plan), and the introduction of Old World animals also brought butter, lard, and tallow.

The traditional pre-Columbian foods are a good match for the cooking techniques. Griddle-cooked tortillas are dry-fried, steamed tamales are steamed, and pozole, atolli, and the variety of "sauces" (stews, soups, sauces) were boiled/simmered or raw.

Worthy of quoting is Sahagun's splendid description of the street food available in marktes, based on tamales and tortillas:

He sells meat tamales; turkey meat packets; plain tamales; tamales cooked in an earth oven; those cooked in an olla...grains of maize with chile, tamales with chile...fish tamales, fish with grains of maize, frog tamales, frog with grains of maize, axolotl with grains of maize, axolotl tamales, tamales with grains of maize, mushrooms with grains of maize, tuna cactus with grains of maize, rabbit tamales, rabbit with grains of maize, pocket gopher tamales: tasty--tasty, very tasty...Where [it is] tasty, [it has] chile, salt, tomates, squash seeds: shredded, crumbled, juiced. He sells tamales of maize softened in wood ashes, the water of tamales, tamales of maize softened in lime--narrow tamales, fruit tamales, cooked bean tamales,; cooked beans with grains of maize, cracked beans with grains of maize; broke, cracked grains of maize. [He sells] salted wide tamales, tamales bound up on top, [with] grains of maize thrown in; crumbled, pounded tamales; spotted tamales, pointed tamales, white fruit tamales, red fruit tamales, turkey egg tamales, turkey eggs with grains of maize; tamales of tender maize, tamales of green maize, brick-shaped tamales, braised ones; plain tamales, honey tamales, bee tamales, tamales with grains of maize, squash tamales, crumbled tamales, maize flower tamales. The bed food seller [is] he who sells filthy tamales, discolored tamales--broken, tasteless, quite tasteless, inedible, frightenting, deceiving; tamales made of chaff, swollen tamales, spoiled tamales, foul tamales--sticky, gummy; old tamales, cold tamales-- dirty and sour, very sour, exceedingly sour, stinking. The food seller sells tortillas which [are] thick, thickish, thick overall, extremely thick; he sells thin [ones]--thin tortillas, stretch-out tortillas,; disclike, straight...with shelled beans, cooked shelled beans, uncooked shelled beans; with shelled beans mahsed; chile with maize, tortillas with meat and grains of maize, folded...with chile--chile wrapped, gathered in the hand; ashen tortillas, washed tortillas. He sells folded tortillas, thick tortillas, coarse tortillas. He sells tortillas with turkey eggs, tortillas made with honey, pressed ones, glove-shaped tortillas, plain tortillas, assorted ones, braised ones, sweet tortillas, amaranth seed tortillas, squash tortillas, green maize tortillas, brick-shaped tortillas, tuna cactus tortillas; broken, crumbled, old tortillas; cold tortillas, toasted ones, dried tortillas, stinking tortillas. He sells foods sauces, hot sauces; fried [food], olla-cooked [food], juices, sauces of juices, shredded [food] with chile, with squash seeds, with tomatoes, with smoked chile, with hot chile, with yellow chile, with mild red chile sauce, yellow chile sauce, hot chile sauce, with "bird excrement" sauce, sauce of smoked chile, heated [sauces], bean sauce; [he sells] toasted beans, cooked beans, mushroom sauce, sauce of small squash, sauce of large tomatoes, sauce of ordinary tomatoes, sauce of various kinds of sour herbs, avocado sauce.

Sahagun, as quoted in Sophie D. Coe, America's First Cuisines, University of Texas Press, 1994 (and copy-and-pasted from here, which also gives other relevant quotes (mostly secondary rather than primary sources)).