The biggest difference that I'm aware of is that the Classical Greek religion was much more the religion of myths that we all know, while the Classical Roman religion had fewer personifications and its gods were much more like numinous forces than like people.
The Greek religion that we know was encapsulated by Homer who served in some respects like an Old Testament to their classical religion. The Greek gods seem to have alwayds been personified -- they were basically powerful -- sometimes very powerful -- people. The Greek religion was by-and-large not about morality (how could any of the Olympians claim superiority there?) but about propitiation of potentially dangerous powers and making deals with potentially friendly ones who might thereafter act as your supporter or your city's patron.
During the course of classical antiquity many philosophers examined Greek religion and often found it wanting. By the height of Greek culture, the religion still existed, but the various philosophical schools -- the Cynics, the Sophists, the Pythagoreans, the Stoics, the Epicureans -- provided a guide to living which the Greek religion lacked. But the Olympians were deeply embedded in Greek culture, and you could certainly get in trouble by being too overt in your questioning of them -- just ask Socrates. But they appear to have been more a matter of social solidarity than what we'd understand as a religion.
The Romans were broadly similar, but the earliest things we know of the Roman gods show them as faceless, unembodied personifications of places and forces: Luck, War, the Sky, Fertility, Crossroads, Family, etc., etc. As the Romans became more aware of Greek culture they came to see their gods as having Greek equivalents and their view of the gods tended towards the Greek model, but never completely. The Roman gods were always numinous forces as much as they were people with great powers.
The Roman religion has been characterizes as being "organized superstition" and some Roman rituals were meaningless to the participants and done only because they'd always been done that way and it might be bad luck to stop. Roman religious rituals also had the feature that they had to be done perfectly with any slightest error requiring the officiant to start from the beginning.
The Roman religion was intimately tied to the Republic, with the officials of the Republic conducting many rituals. There were priests, but they were almost invariably powerful officials who were elected or appointed to the post and who used their religious position to further their political one. (Famously, Julius Caesar's political career was saved when he manged to get elected Pontifex Maximus.)
As the Republic turned into the Empire, the Roman religion was even more co-opted for political ends, and the worship of deified emperors and of Rome itself became an increasing focus, not because people believed that Augustus now sat on Mt Olympus or wherever, but because it provided a unifying force in a giant organization that badly needed them. (This was a large part of why Christians were persecuted: they didn't participate in civic rituals that the State considered important, but (because they were new and didn't have a long history) lacked the Get Out of Sacrifice Free card that the Romans (usually) gave the Jews.)
With the Roman religion providing as little spiritual guidance or solace as the Greek, the Greek philosophical schools were also heavily adopted by the Romans.
Bottom line: The Greeks tended towards greater personification of their gods; the Romans tended towards their religion being a series of quid pro quo transactions with faceless forces. But they had much in common and more as time went on and the cultures merged.
Edit:
Off-hand, one book I found very interesting was Religions of Rome: Vol 1 – A History by Mary Beard, John North, and Simon Price, Cambridge, 978-0-521-31682-8, 454 pp, 1998.
