Both the Romans and the Aztecs can be described as people who linked together trade networks and incorporated millions of people of various cultural backgrounds. This however does not remove the fact that they were capable of great violence and expanded their domains through conquest. You can compare the two in a number of ways in fact.

Looking at the beginnings of both states, they both originated as small city-states that were a part of regions (Central Mexico and Ancient Italy) that knew conflict that ranged from organized warfare to seasonal raids. This helped shape both the Aztecs and Romans into martial societies. The nobility/leadership of both peoples had their political fortunes tied to military accomplishments, so warfare became more than just for national defense or conquest but for political success as well. In Aztec society nobles were groomed for command, priesthood and administrative functions while commoners were given a military education and one of their few chances for social advancement was through military feats and accomplishments. In Rome the so called cursus honorum, traditional path to political offices, began with military service, while later service in the military guaranteed foreign peoples citizenship. Hopefully this provides a clue to how the militaries of the states played important roles.

The other comparison point I want to make is how both states expanded. While many view Rome as a monolithic empire, the reality is that for much of its Republican period, Rome was a patchwork of alliances. In a typical episode Rome would offer a subjugated people an alliance in which Rome was the senior partner. They would force the losing party to cede a plot of land for a Roman settlement, they would have to provide Rome with soldiers during times of conflict, and tribute could be extracted. In return the subjugated people were able to engage in commercial activities with Rome, make use of Roman infrastructure (i.e. roads), and have Rome's protection. Similarly, the Triple Alliance didn't annex people, but instead forced them to acknowledge their superiority and provide troops, tribute, and hostages. They too benefited from the stability and economic pluses from the Aztec umbrella. The greatest difference between the two was that Rome was able to "Romanize" foreign subjects to a greater degree because of its longevity and because its institutions were geared to it.

Hopefully all these words reinforced what I said: that both were martial societies that managed to capitalize on their conquests by incorporating subjugated people into their empires to various degrees of success. The regions that they consolidated would benefit from the stability that they provided and the commerce that they stimulated. Two say one was more peaceful than the other is semantics. You don't go from a small settlement to an empire by playing nicely.

Edits for grammar, it was late.

The Aztecs weren't the inhumane demons the chroniclers often painted them as, but neither was the Triple Alliance particularly nice either. The borders of the empire were the subject of the greatest domination and force, to hedge against the encroachment of hostile polities. The Toluca Valley was an especially dominated region. Michael Smith has a broad overview of his work in the region in [1], but more recent excavations under Angela Huster at Matlatzinco have recorded significant drops in the material wealth of households as well as a decrease in the size of elite residences immediately following Aztec conquest.

[1] Tomaszewski, B. M., & Smith, M. E. (2011). Polities, territory and historical change in Postclassic Matlatzinco (Toluca Valley, central Mexico). Journal of Historical Geography, 37(1), 22-39.