This is something Mary Beard has covered in a lot of detail in her books that I really appreciate her focus on, even if Roman history isn't really my specialty. Essentially, in a lot of cases the dichotomy between gods and mortals was not nearly as clear for ancient Roman paganism than it is in the abrahamic conception of gods. It was a permeable barrier where the powers of an emperor to rule the known world could walk right up to the border of the ancient conception of godhood. At the same time though, Romans also did not look at mortal people as exactly alike to the Olympian gods. For their actual lifetimes most of the people given divine honors were treated far more like a devout Catholic would treat the saints. Prayers and sacrifices weren't really for them but in honor of them with a more hazy conception of what it then meant spiritually. After their deaths though they were in the very least regarded as in the same rough category as many of the minor gods of their religion. But at the same time, we have enough satirical works of Roman comedians mocking and making light of the standing of many of these gods to say their honors could also often be purely formal. I personally doubt anyone took the deification of Antoninus as anything more than the emperor making an extreme statement of grief at his loss. So in general there could be a lot of crossover with simply heavily memorializing someone, with the goal of keeping their name and memory alive. That's almost certainly the case with someone like Cicero when he was contemplating building a temple in honor of his daughter who died in childbirth who he always deeply loved.

At the same time though this shouldn't be taken to mean that they weren't given huge honors and treated with genuine reverence comparable to the gods. They were, especially figures like Augustus who loomed large in the memory of the average person and could even take on a similar level of veneration as any of the Olympic gods. We know that there were active priesthoods of Augustus spread out across the Roman world. But that does mean that a lot depended on the way the emperor or important person was remembered after their death.

It should also be said as a final note that worship of any singular Roman God would have never really risen to the level of a religion in itself. Romans did essentially worship all of the gods in some measure, even if they might have favorites, or integrate various local and foreign deities into their religious observations. The religion itself is more the entire system of beliefs surrounding the gods, who they were, and what they meant for the world than worship of any specific gods.