What was the role of religion in Greece and the Roman Empire?
Of course, Christianity was FOUNDED in the Roman imperial period and became the official religion of the Roman Empire under Constantine, so, if one wanted to be accurate, one would not contrast Christianity with the religions of the Roman "empire".
Instead, one could say that the "do ut des" model of pagan Greek and Roman religion, focused primarily on ritual rather than theology, applied mainly to the official state religions (before Constantine), but in the wide and diverse cultures which included areas now covered by some 13 modern countries over the period from the death of Alexander to the fall of the western Roman empire (which is what is correctly termed "empire"), there were many diverse models of religion including salvation religions, mystery religions, etc. Among the religions of the period and place were Judaism, Christianity, Manicheism, gnosticism, Mithraism, the Elysian and other myster religions, etc.
The role of religion in these classical societies was more practical and social than it was personal. These were not really societies in which people felt personal relationships with the gods in the way that we Christians are supposed to feel a connection to Jesus/God. Instead, religion in these classical societies was more of a ritual function that had to be fulifilled. People believed that performing the correct rituals in the correct ways would bring success to their country. They also believed that they could determine the will of the gods through things like omens. In these ways, it was much more of a practical relationship in which people did what the gods wanted in exchange for certain benefits. It was not a personal and emotional religion as religion has come to be in the modern world.