Well, if we're going to throw souled!Spike into a discussion about Liam/Angelus/Angel, then I think we need to consider the way those two work together as foils. They reflect and highlight various aspects of each other's personalities. This setup is pretty common in mythology and literature from many different cultures and time periods. Think about Jacob and Essau, Castor and Pollux, the Ashvin brothers, Hamlet and Laertes, Sherlock Holmes and Dr. Watson, etc. Spike is very much defined in relation to Angel, so talking about what Spike is -- and what he isn't -- should help us to better define Angel.
Because I'm a Jane Austen nut, I'm gonna bring up Wickham and Darcy. In Pride and Prejudice, the hero is a very reserved character. He's also not the main focus of the novel and spends much of the novel "off stage," so to speak. So how do we come to understand and root for the hero? We examine Wickham, his foil. By getting to know both men separately and comparing their opposite manners and actions, the heroine of the novel is able to sketch their characters in order to decide which is the good man and which only has the appearance of goodness. By this process, the hero is revealed as such even when he is not present for the current scene. Wickham and Darcy represent two very different models of Regency era manhood: the eighteenth century Chesterfieldian model and the emerging Victorian model. In much the same way, Angel and Spike represent those same two models, but of souled vampirism (or even heroism).
So let's look at how these boys are the same. They're from the same vampire family. They have an icky habit of comshucking (and, arguably, even loving) some of the same women. They are both famous and much-feared killers. They are both souled vampires. They are both world-saving champions. They were both sired by mother/lover figures. When first souled, they both suffer madness (of an apparently none-too-hygienic type) and crushing guilt. They both want the same two things: Buffy and redemption. Also, neither one really seems to believe they will get those things in the end, so with nothing to really risk or win, they both tend to display a willingness to sacrifice themselves for the good of the world. Souled, they both also display a tendency toward doing the right thing just because it is right. Killing demons is one thing, but with souls, both find the idea of killing humans extremely distressing. Being souled doesn't just put a muzzle on their aggressive, bloodthirsty urges (like the chip did). Instead, having a soul takes away those urges (or at least lessens them to normal human levels). Basically, they serve the same essential function within the Buffyverse. That means that the differences have to be pretty darn meaningful.
So how are they different? This should prove enlightening. They differ in age, with Angel being the elder (and, therefore, usually the preferred) of the two. Spike often plays the underdog. Along with the subject of age, we should mention that Angel (as Liam) was reared in the Age of Reason, whereas Spike was reared in a veritable Age of Repression. They differ in nationality, with Spike's nation being the conquerors of Angel's nation. They differ greatly in physical appearance, with Angel being dark and having a bulky, traditionally manly physique. Spike is more lightly colored and has a compact, angular, more effeminate appearance (especially in early seasons, with the eyeliner and the painted nails). They were different kinds of evil, with Angel (as Angelus) glorying in slow psychological torment and Spike liking the adrenaline rush of a good, fast fight. Spike's driving motivation has always been love, whereas Angel's was his desire to be the best (whether he wanted to be the best killer or the best champion). The acquisition of their souls was very different. Angel's was foisted upon him as a curse, whereas Spike actively fought for his. Souled, Angel's personality fragmented from two to three distinct counterparts. Conversely, Spike's transition is less overt and dramatic. Angelus is cold and calculating when Angel would rely on bravery and purity of intention. Souled!Spike, however, is still Mr. Fists-and-Fangs.
Based on these similarities and differences, I think we can make some conclusions about how a soul does and doesn't function within a souled vampire. First and foremost, it diminishes aggressive, selfish impulses to human (or near human) levels. It promotes selflessness. It also acts as a conscience, inducing extreme amounts of guilt. The differences tell us what it doesn't do. If the soul changes something in Angel and not in Spike, then we can probably say that effect was caused by another variable, right? So what it doesn't do is alter appearance, personality, motivation, fighting style, etc. The soul doesn't change who someone is. It doesn't create two separate people. It just takes away the baser, unchecked vampire urges and replaces them with more heroic ones.
That means that, in vampires, the soul appears to affect the amygdala, which is the section of the brain that controls aggression, eating, drinking, and sexual behaviors. That makes sense, since vampirism is often used as a metaphor for sex and since the major areas of difference between vampires and humans are displayed in their appetites. Without the soul, the amygdala is hyperactive. With the soul, this area of the brain is suppressed. The person is in every other way unchanged except as that person would normally change due to experience or other factors. Souled or unsouled, Spike and Angel retain very real and striking differences of character, personality, and motivation. If the soul did really change personhood, Spike would have displayed a "break" in personalities when his soul was restored. Based on this, I would argue that Angel is not a different person from Angelus. Instead, the apparent "break" in personalities is probably a psychiatric condition brought about by external (or internal) variables. That also coincides with my theory of Angel as a Freudian figure.
So, yeah. I think souled or unsouled, Angel is the same person. The fractured personality is not the result of the soul but, rather, the result of some other variable or set of variables. The BIG difference, of course, is the way the two vampires obtained their souls. It seems possible, then, that Angel's fractured personality is the result of the trauma of having a soul suddenly forced upon him, whereas Spike's more ready integration of the soul is the result of the voluntary nature of his re-ensoulment as well as the prolonged period of "preparation" resulting from his time with the chip and his corresponding reintegration into human society. I don't think it's the soul itself that fractured Angel's personality. Rather, I think it was the method and timeline of reinserting the soul that did it. It's a subtle, yet important difference.
