I can certainly answer how it changed over time. Namely, very intriguing details from the 2nd century onwards have shown evidence of growing Monotheism in the Roman world via Jupiter/Zeus or Heracles, bereft of very little if any intervention from Judeo Christian influences. This indicates that over time, the Greco-Roman world wanted a more stable and centralized religious system that the traditional pagan systems certainly did not offer in any way, shape or form due to geopolitical and cultural reasons. In centuries past, each city-state had their own deities that protected them while demanding worship in turn. Rome had Mars/Ares, Delphi had Apollo, Athens had Athena, and so on. They were seen as essential as a form of nationalism/tribalism which represented their ethos and what they wanted to stand for. A myth of Athens' name came from when Poseidon and Athena were fighting over who would become a patron of the city. She eventually outwitted him and he was denied. Romulus and Remus were the sons of Mars and the Romans descendants of the Trojans in the Aeneid, who came from a city that primarily worshipped Apollo.

However, this since gradually became obsolete due to geopolitical changes. A city state or a league of city states was no longer a tenable system or hadn’t been for a while. The Trojan War for instance was filled with competing interests and an incoherent leadership, solely dependent on the hopes that every city-state would support the interests of Greece as a whole. In contrast, the Persian not only had one single religion at the top, but a coherent and much simpler ideology that didn't need to step on eggshells in order to fulfill basic worship rites like the Greco-Roman model. We see in virtually every war the Greeks and the Persians had how this model was a hindrance to any sort of stability. The Macedonians’ rise alongside the eventual rise of Rome diminished these city-states because they were simply a hindrance to any geopolitical or socioeconomic wellbeing that needed to be established in order to make a stable power.

It seems rather obvious that people would get tired of the Greco-Roman system in how it treated people, down to the horrid afterlife and the sheer nonsense that precluded itself. The worship of other gods gradually declined from the Emperor being the Pontifex Maximus as well as the secular leader of the Empire and thus, the representative of Jupiter on Earth. Despite this, inspiration from the Judeo-Christian system gave some Romans and Greeks realization that a coherent monotheism would do much better for them and the Empire as a whole than dozens of gods that no longer offered any single utilitarian use towards political or cultural matters. It comes to my attention that some of the lesser gods that weren’t Jupiter/Zeus waned in popularity during this period including some of the pagan temples The original Olympics degraded over those centuries as well due to this shift of power until they were rendered irrelevant until they were eventually banned by Theodosius. With waning popularity leads to loss of funds, underhanded dealings and eventually closure.

We can see the effect of demographic shifts on the closure of [Catholic] churches and schools in the United States as well as in Western Europe. I placed Catholic in brackets because it is not mutually exclusive in Western Europe due to Protestant churches. Those churches closed from lower demand of worshiping places alongside religious shifts away from organized religion and noted controversies associated with said sects. In the same respect, pagan temples grew out of use due to a simple lack of demand and revulsion for the older ways that a reformed Jupiterism, Heracleanity ,Judaism, Christianity, or Mithraism would offer in spades. However, I am aware the Christianity, though from an emperor’s eyes a decent basis for societal stabilization, was very complicated in the earlier centuries as it had various sects and interpretations of its own which needed to be sorted out. The Arian and Coptic sects are both rather intriguing instances and their political relationship with the Orthodox Roman/Byzantine Empire’s hegemony. Alexandria and Rome’s political reactions to Constantinople’s arrogance parallel how the Ancient Greco-Roman city-states acted to a certain extent. Rather than give in, the Copts accepted Islamic rule while the Romans found authority in the Franks as a stabilizing force [that Constantinople couldn’t or wouldn’t provide either.] However, unlike the Greeks, geopolitical instances where an effectively permanent reconquista of formerly Christian lands never occurred akin to an Alexander the Great or an Ottoman dynasty that simply converted to Orthodoxy very early in its existence.

And of course, we cannot forget about the fallout of the Protestant Reformation illustrated the necessity and convenience for local North European powers to break away from Rome either. Though the break up itself, while violent and traumatic, isn’t quite as devastating in relations as the 1054 schism and the subsequent 1204 Sack of Constantinople where we ultimately find the Byzantines preferring Ottoman rule. For another modern comparison, there were many football teams in the United States in the early days of the sport, however today, there is usually now only a single professional team per state. And for most sports, the teams decrease up to the Olympic level, where there is only one national team. We do not see minor league teams compete in the same league as Major League Baseball or college to the NFL, that adherents to the minor Olympian gods would compete with the almighty Zeus. Sports adherents tend to act in similar ways as the Greco Roman pagans and other sectarians up to the fanatical worship of the locality that divides them from the rest of the country itself.

To conclude, Greco-Roman religion went from a system of tribalistic polytheism to a gradual simplification towards monotheism due to geopolitical and cultural necessities that tribalistic polytheism could never offer.