I've come across this sentence, and I'm not sure the effect of 'on' here:
Every smoker is an embodiment of Prometheus, stealing fire from gods and bringing it on back home.
Without 'on', would it be a major difference in terms of meaning?
That sentence means the same thing with or without "on", but feels much more informal and familiar with "on", and is almost never used in writing.
The word "on" can be inserted like this before the particle (the second part) of many phrasal verbs:
bring it on back
cheer someone on up
drink it on up
come on by
drop on out
lie on down
and so on.
Some varieties of English allow this more than others. For example, in most (all?) varieties of English, people naturally say, "come on by", but "cheer on up" sounds like someone from Texas or the Midwest US, and as far as I know, nobody ever says, "hold on on".
In this specific example, it's purposefully used to make "bringing it back home" contrast sharply in tone with "Every smoker is an embodiment of Prometheus, stealing fire from gods", which sounds highly formal. This makes it even clearer that when a smoker steals fire from the gods, it really belongs to regular, informal and friendly people, and no longer to the heavenly, formal and brutal gods.
Adding "on" to "bring" does not change the meaning, but it sounds more informal.
For example, "Bring the paper down to the office" means the same thing as "Bring the paper on down to the office," but the latter sounds significantly less formal. It works the same way in the sentence you posted.
I would say that adding "on" to "bring" is also somewhat more common in southern-US dialects.