The differences between the two are greater and more significant than the similarities. In fact, the Aztecs probably bear a greater resemblance to Sparta than to Rome. It should be noted first that the Aztec Empire was relatively short-lived; existing scarcely two hundred years; whereas the Western Roman Republic and Empire alone was extant for almost one thousand years.
Aztecs called themselves Mexica; and early on had a reputation for causing trouble, often seizing land cultivated by others. They were chased away early on, but later settled and built their capital city of Tenochtitlan on an island. Aztec society was exceptionally militant. All males were considered potential warriors, and the entire hierarchy of the culture was based on militarism. They imposed a heavy tribute on subject peoples, often enforced by brutal military reprisals if there were rebellions. The tribute was used to make the Aztec elite quite wealthy. A portion of the tribute consisted of sacrificial victims for the Aztec gods, who demanded a constant supply of human hearts. So brutal were the Aztecs that it is doubtful Hernan Cortes could have overcome them had he not been joined by tribes subjected by the Aztecs who were anxious to be rid of them.
In contrast, Roman society developed when Junius Brutus overthrew a brutal Etruscan ruler. Ultimately the Roman Empire occupied a far larger area and lasted much longer than the Aztec. A substantial factor in this difference is that the Romans tended to quite tolerant of conquered peoples whereas the Aztecs were brutal and vindictive. The Romans were quite tolerant of religious differences, did not require subject people to pay taxes, although they did have to furnish soldiers for the Roman army, and only required that they not ally with the enemies of Rome.
The contrast then is between a militant society more similar to Sparta than any other European power and a cosmopolitan society which tolerated diversity.