I'm just going to go off with it here in the hopes that I can write quickly for a change, but maybe I can clear it up through discussion afterwards. Note first of all that I'm talking in Buffyverse terms only, and this has nothing to do with my beliefs on any other topic.
(All this Buffy stuff lately is like a blast from the past! I probably meant to make a post like this around four years ago.)
There are two ways that a human body can be animated. First is by the soul, or in other words, being alive. Soul plus body come together to make a human, which is the automatic process of our life cycle. Death means the soul and body have been separated and there's nothing to animate the body.
Second is with a demon. Demons come from some other dimension I suppose. A vampire makes a new vampire by killing a human - i.e., removing a soul - and through the exchange of blood, summons up a new demon to take over and animate the body. There are a few telltale signs that this is not natural: the body is still dead, and the demon doesn't act like a soul.
Where the soul goes after death is up for debate. Personally, I favor taaroko's theory that it can't move on to the afterlife while the demon is animating the body, so it's essentially trapped in limbo until the vampire is dusted.
In Angel's case, the demon takeover had been completed as normal and Angelus was a standard demon animating a standard human body. When he was cursed, his soul was pulled back from wherever it had been back into the body. The important thing to note here is that it didn't and can't replace the demon. Liam's body is dead and a soul can only animate the body it's born into through the duration of that body's life.
So he's got both. The demon is still doing its thing, keeping his body in immortal stasis, requiring blood, and craving violence and evil deeds. What's the soul doing? Fighting the demon.
The tricky part to understanding this is the way "demon and soul in one body, fighting for dominance" ends up sounding like a dual personality. Does Angel have conversations with Angelus in his head? Canon doesn't support that, and it sounds kind of silly anyway.
However, there's one more element to consider: the mind. You can look at this as an extension of the body (and indeed, it comes from brainwaves, which are biological). Neither a demon nor a soul has a mind of its own. A human soul takes the character of the body/mind it inhabits, growing with experience and eventually leaning toward good or evil impulses. A demon knocks the soul out, but keeps the mind. That's why vampires keep the personality and mannerisms of the human body, but not the moral alignment.
Some think that the variation in how vampires imitate their human predecessors has to do with the relative strength of the demon that took over, or possibly the strength of the soul that it dislodged. I don't. I think demons are all essentially identical, and Angel, again, wasn't a special case until he got hit with the curse. The reason that Angelus is so unlike Liam is all about the mind - the personality that he had developed as an unhappy, rebellious young man was one that adapted splendidly to a life of power and revenge without guilt.
Like the soul and the body, the soul and the mind aren't meant to be separated. When they are, it's an instant identity crisis: Angel's kept his mind throughout the entire ordeal, including all of the memories and all of the emotions that went with them. As far as he can tell, it's been him the whole time. He, Liam, had the ability to make good choices, but didn't. He, Angelus, was only capable of evil and liked it that way. Now he, Angel, has the baggage from both of them.
His current personality is the product of three things. First, it's the same way the personality develops for any of us, albeit with a longer time span and much weirder twists - everything he's done and experienced has affected him in some way. Second, it's the demon in his brain, who wants blood, sex, the suffering of others, and precious little else, and it wants them bad. Third, it's his soul. It's not a remarkable soul, and it's not even required for keeping his body alive, so there really isn't much that it can do except counteract the pain of his history and the desires of his demon with the feeling of responsibility to do the right thing, and it's hopelessly outmatched.
The reason I adore Angel is that it's still winning.
When we see Angelus again it's a clear indicator of how much work his soul is doing as Angel. He's reduced once again to body + mind + demon, and the absence makes all the difference in the world. He knows everything about Sunnydale's supernatural activity and uses it to terrorize the population instead of protecting it. He remembers how it felt to love Buffy, and he despises her for it. He's even got the same intuitive understanding of people, but instead of using it to know who to trust, he uses it to manipulate and torture them. Same guy, but evil - and he's only evil because nothing is there to keep it in check.
Contrast to Spike, who gets a soul added to his mind + demon and goes on much the same as he did before: apparently, his demon gets along pretty well with his soul already, so it's likely dictated mostly by the mind. Or Harmony, who barely changes once she loses her soul and gets her demon. Her soul wasn't worth much.
So, Angel and Angelus aren't exactly the same person. But wherever Angel is, Angelus is too, and though we may not see him making any appearances, Angel's feeling him and he's working hard to make sure nobody else does. When the curse breaks, though, the soul's just gone and nothing's getting in Angelus's way. That's why he doesn't brood.
Although Buffy isn't the type to lay out an analysis like this and base her decisions on it, I think she has an inherent understanding of it based on her experience in Season 2. She fell in love with Angel as Angel, vampire-with-a-soul, and although she can accept the evil side of him which has always been there, she can't accept him without a soul. That's the core of the real Angel, and valuing anything over it is superficial, especially once she sees how extraordinary it is that he's facing these incredible odds every day and coming out on top. Buffy knows what constant struggle is. It speaks to her more than personality.
I'm just going to go ahead and post this without proofreading it. I never do that, by the way, so if this is the first meta I've written that doesn't make any sense, now you know why.