I always thought the shows [Both Buffy and Angel] missed a golden opportunity by never delving into a frank debate about the nature of souls [in the context of the show] and Vampires. To me it always seemedlike the elephant in the room that no one talked about, and while I did enjoy both shows to varying degrees despite their flaws, this failure of omission always seemed particularly striking.
Personally, I’ve never bought the “Vampires, demons, etc” have no soul line. It works for a quick ‘on-off’ but it makes no sense in the context of a consistent created world.
Therefore, I came up with my own rationalization so I would stop being annoyed and just enjoy the show for what it’s worth.
First off, Demons:
Demons have a soul. And just like humans and demons are different physically so are their souls, and thus, their properties and characteristics are sufficiently different to be unrecognizable to human scholars who don’t believe demons have souls in the first place and so don’t look very hard.
The fact that demons don’t reference their souls in the context of the show is immaterial. First, we don’t spend a lot of time following the lives and struggles of demons so we don’t know if the issue ever comes up. Second, given that demons cultures are all different from human [as well as from each other] it’s certainly possible that the concept of what a soul -is- may be separate. And third, it may be that demon souls are typically fused closer to their physical forms then their human counterparts and thus they feel no need to generally make a distinction. This last one would make demon to demon possession difficult which might be why we don’t see a lot of demon body swaps/soul transfers in the show.
Nextf, Vampires:
Vampires have a soul. Or, more correctly, they are a soul. A a nasty vicious demon Soul. Back in the day maybe they had more but that was stripped away when all the big capital D Demons left. They had a will to stay and managed to somehow skip the exodus by separating out their physical forms and their souls, possessing humans to replace the form physical form they lost and more or less fooling whomever organized the big ‘bye-bye’ [all the big demons and gods leaving is referenced but never really made clear in the show, which works fine for me].
Thus, the first vampire[s] was born, a new sort of demon. These demons, known as vampires, reproduce by ripping out a bit of their own soul and inserting it into a new form like a virus or parasite. The little bit of vampire soul then enters/infects the human host and like a parasite, drains power [will, soulstuff whatever] from the human’s soul to build itself. Thus, the demon soul becomes stronger by taking from the human soul and eventually takes over. Generally the human soul is not completely consumed but there is so little left that it unrecognizable and so little strength that it has little influence. Whereas human souls seem to sit idly in their bodies like cookies in a jar, being a demon soul it fuses itself into the physical form [possibly as an anchor to maintain its hold] binding the two together, which is why vampires turn to dust when they die.
The fact that Vampires don’t seem to suffer any long term ill-effects would tend to indicate that over time their souls are capable of replenishing themselves somewhat, possibly through feeding.
Now, when Angel gets his soul back, all of a sudden there is a fully formed human soul in the body along with the demon. However, at this point the demon is no longer in its pupae or parasite stage but fully incorporated with the body and can't pull the same trick before. This allows the human soul to seize control of the physical form, demon fused bits and all. The demon never goes away of course, but unless something comes along to shake things up, the demon won't be able to take control back. This, "shaking things up" can be anything from a spell, another demon, curse, whatever. Anything to jar or weaken the human soul sufficiently.
Anyway, that’s my explanation for demons, vampires and souls in the buffyverse. I think it covers most eventualities fairly well along with explaining most of what occurs in the canon.