The question puts in contrasts Mayans and Aztecs with Romans, and all these were developed civilizations, where agriculture was not at all a recent discovery. Another example that could be mentioned is that of the Carthaginians, who practiced human (especially child) sacrifice. But each case may need to be considered separately, as it might prove problematic to assign a common single cause to all cases.
Maybe the answer could be given in terms of history of religion, and also anthropology and philosophy of religion, rather than in simply "historical" terms, considering:
- the place and logic of religion within each of the aforementioned societies (which may be considered more or less "religious")
- the place of human sacrifice within each of the religions of these societies (as different religions may prove to be more or less "sacrificial").
Each of these points are problematic by themselves, as they depend on very different answers that can be given to questions like:
Is human sacrifice an "essential" aspect of the religious phenomenon?
Is human sacrifice a "primitive" trend in the evolution of religions and of social institutions?
Is therefore the disappearance or rarity of human sacrifice the sign or the effect of religion practice and institutions being superceded by newer institutions with a different/separate logic - namely the political ones? Why is that not the case with the Aztecs then?
As such arguments are more philosophical than historical/archaeological, they are all controversial, but they may help provide a few basic terms with which an answer could be formulated.
For example, on the place of sacrifice in the evolution of religion, an interesting theory is that of René Girard; he considers religion as the first real human institution and as the capital means through which humans have come to terms with intestine violence. Girard considers intestine (or "mimetic") violence an essential aspect of human societies and sacrifice as the primitive way to control, re-direct and so to speak "domesticate" the violence of human societies. Also, he sees an anti-sacrificial trend in the evolution of both religion and human societies. Religion and violence became separated, violence came to be seen as anti-religious or as exceptional.
On the way "primitive" religion is gradually replaced by a religion that is controlled by political logic and institutions, a very philosophical (in hegelian terms) but interesting book is The Disenchantment of the World by Marcel Gauchet. It analyzes how political power has its own logic that gradually replaces "religious" logic, how the city states and the first empires impose changes of the religion itself, and of the very meaning of religion. Gauchet is not interested in the violent aspect of religion but the two theories can be used together in order to understand how elements that initially were mingled together (sacred, violence, power) gradually became differentiated.
A different theory on religion, that of Mircea Eliade, confirms to a point this perspective. When he considers the "essence" of religion, namely the "sacred", he understands it in philosophical/ontological terms, far from the violent/sacrificial aspect of Girard's theory; but for Eliade, like for Girard, to understand the "ground zero" of religion and the "religious" in its purest form is to go back to the most primitive forms of religion. And when Eliade analyzes the religions of Romans and Greeks (and of other city-states and empires) he points out the fact that the most primitive aspects have disappeared to a great extent or have been re-interpreted in a context that (in this view) is less "religious". This confirms the perspective described above, where we have an evolution of religion in relation to institutions that gradually free themselves from the strictly religious and impose a new world with new meanings (what Gauchet called "the Disenchantment of the World" and Eliade calls desacralization).
These new meanings have gradually started to include more and more aspects of what we now see as politics, justice, philosophy and science, while keeping a lot of religious "packaging" all the way.
Trying to articulate an answer to the question:
First, as I said, the question is not about "primitive" religion (where there are no states, powerful armies, etc), but about the religion of rather developed civilizations. Whether sacrifice is a "primitive" trend of religion may be a matter of speculation but, in the context of this question it may be considered so . This is confirmed by the answers that point out the relation between primitive agriculture and human sacrifices. In this sense, for the sake of this argument, human sacrifice will be considered "normal" at this "primitive" level, but problematic at a later date.
The question, then, amounts to asking how come a such "primitive" aspect is present in some developed civilizations and is absent in others.
It is noticeable that the disappearance of human sacrifice means less violent religions, but it doesn't mean less violent societies, on the contrary. Violence becomes the monopoly of far more powerful institutions than the strictly religious ones. These institutions are the political ones; they have a bigger control over intestine violence but are able to exercise a greater violence than ever before, either internally (as violent authority) or externally (as conquering kingdoms and empires).
As for the answer, I think we have to consider the case of those civilizations where human sacrifice has disappeared. Why has it disappeared? As sacrificial violence tends to melt away into judicial violence, the development of powerful justice practices and institutions contributed to the full abandonment of human sacrifice.
In the case of the Romans, Greeks and many other ancient civilizations, the sacrificial aspects of religion, especially human sacrifice, have greatly receded, and have become not only rare, but exceptional, or completely absent. Animal sacrifice has thus become the rule. An odd case is that of the Athenian pharmakos (French link is more detailed); it may have been rare, and the institution itself, without being exceptional, triggered the sacrificial event only in exceptional situations of crisis. The logic of such sacrifices is that when the "modern" institutions failed (in cases of military or natural disasters), the "old" mechanisms were used to help improve the situation.
A somewhat different case is that of civilizations where human sacrifice was rare, but not exceptional, as it was done on regular basis at certain moments and places. That seems to be the case up to a point with the Carthaginians, Phoenicians, and other Middle East kingdoms and empires. Retainers sacrifices have been regular up to a certain date in Egypt and China. India's economic, political and religious diversity amounts to a more complex case, because "primitive" human sacrifice (in certain isolated agricultural communities), and other regular sacrifices like that of wifes of deceased or those to Sakti are documented until much recent dates.
A special case is that of the Amerindian empires, and especially of the Aztecs. It seems that in this case the religious institutions related to human sacrifice have not only been kept with the strengthening of the political institutions, but have been further developed and strengthened. What arguably happened was a sort of innovation where sacrificial religion and politics became unified under a single imperialistic ideology, by which the development of the empire stimulated sacrificial religious practices and these practices stimulated the imperial development (as it imposed a constant need for new captives to be sacrificed).
An aspect that is not mentioned in the question is the relation between human sacrifice and canibalism. Canibalism may possibly be the most "primitive" aspect of this "primitive" phenomenon, but this qualification may be controversial. What is not controversial is the fact that Aztecs were also practicing canibalism. This may have been encouraged by the originality of the Aztec religious ideology and by other cultural trends relating to the ecological or biological background. (These ecological reasons are also controversial, but on the importance of biological and climatic circumstances one can find interesting Jared Diamond's book Guns, Germs, and Steel.)
Summing it up: human sacrifice has disappeared due to political and ideological developments that have replaced the role of religious violence with a new religious, political, judicial and military order. This trend was both expressed and encouraged by a second factor, the developments of the religious discourse itself, which became explicitly anti-sacrificial: in some Greek myths, in the Jewish religion, in the Persian religion, in Hinduism and Buddhism. The presence of human sacrifice in certain cases can be explained by the lower presence of one or both of these two factors and/or by a contrary development (the Aztec case) where increasingly powerful political institutions have stimulated sacrificial practices instead of replacing them.