Excellent! I love this topic. I'll hit you with the latest edition of my theory (which is long and has lots of parts).
Before I start with the theory, though, I'm going to mention a couple of things I think are important to establish. First, if you look at the text of Angel's curse, it's made fairly plain that the soul the Romani were giving him was the same soul that had originally belonged to him as Liam (therefore, like Kean said, Angel is essentially Liam+vampiric nature and instincts+150 years of Angelus's baggage). Second, I think one of the most telling things about all the Liam flashbacks is the way he takes the time to attempt to reassure his little sister. That one interaction proves to me that despite the drinking, bar fights, and debauchery, he was not at a cruel man.
Okay, theory now. Here's where I start to make heavy use of the Freudian Split (superego/ego/id), as it fits so well. For the purposes of the Buffyverse, the soul can be equated to the superego, the mind to the ego, and the body to the id. The soul/superego would include morally good traits such as the capacity for selflessness, remorse, love, etc. The mind/ego (which is entirely morally neutral) would include things like memory, intelligence, individual skills (such as fighting ability, artistic talent, creativity, etc.), and possibly the desire for knowledge. The body/id would include things like the physical body itself, as well as selfish human desires (such as the seven deadly sins).
Liam, as a human, has all three (superego, ego, and id). His id tends to influence his ego much more successfully than his superego does, which is why he has surrendered to his vices (which seem to include lust, sloth, and pride in particular).
Angelus, being soulless, lacks the superego, but he isn't just ego and id. (The closest example we have of only ego and id would be Buffy in "Living Conditions", after Kathy had mostly removed her soul.) Vampires are demons. Therefore, something else is in there to keep the body from aging or decaying, which gives it bloodlust and superhuman strength and senses. So let's just call that something else the demon. The demon differs from the id in that instead of being merely selfish, it is actually evil. So Angelus is ego, id, and demon, with the demon clearly being the most dominant of the three.
Angel is all four, with superego dominating and demon coming in second (well, most of the time, anyway). He rarely struggles with his human vices to the same degree that he struggles against his demon.
Human Angel (as seen in IWRY) is back to being superego, ego, and id, with no demon.
Post-hell Angel and Pylea-monster Angel are both superego, id, and demon but with an extremely diminished ego. His mind had been all but destroyed in hell, reducing him to something that is more beast than being. In Pylea, going into vampface renders him unintelligent, but he can still respond positively to Fred. In both cases, without the ego there to serve as the balance for the other parts, he wis left very unstable and without proper control over his actions--just like regular animals (and, possibly, werewolves when they're in their wolf forms).
Every time two characters switch bodies in the shows, they're switching both superego and ego, and perhaps even the non-physical part of the id.
The theory wouldn't work if it didn't also apply to Spike. Happily enough, it does. William is dominated more by his superego (but, unfortunately for him, rather than that making him a wonderful and admirable person, it just sort of makes him pathetic and insecure). The romanticizing sensibilities and intuitiveness are part of the mind, so he keeps them after he turns, though they become twisted in the absence of the superego and presence of the demon.
As a soulless vampire, Spike is dominated by his id more than his demon. He's more about having fun than being evil. Thanks to the demon, though, "fun" and "evil" are fairly synonymous. Over the years of being chipped, he has no choice but to attempt to suppress the demon in order to avoid crippling pain, which means the id gets even more control. The id may be human, but it is selfish. Spike's good actions either have selfish motivations (such as self-preservation and scoring points with Buffy) or are the result of those romantic sensibilities flaring up. He cannot do good out of a genuine capacity for doing selfless good, otherwise he a) breaks the rules of how vampires work and b) is extra horrible for still doing all that evil despite having a capacity for good.
Soulful Spike is still very dominated by his id, though the superego makes its presence clearly known (sometimes). This is why Spike often seems the same, soul or no soul (but not a whole lot like William either way), whereas Angel and Angelus seem like completely different people.
Okay, here's another important piece of the theory that ties it more tightly to the Buffyverse. The soul and the mind are meant to stay together as a seamless unit. The mind gives the soul expression, thought, and awareness. When a human dies, normally both the soul and mind vacate the body together. This would explain why Buffy said she was still herself while she was in heaven, but had no form (because she'd need her body for that). However, when a human is turned into a vampire, only the soul leaves (which is why all of the vampires who regain their souls only have one set of memories, because memories are part of the mind's domain). The mind remains in the body (which is why a vampire is intelligent, where, for instance, a zombie is not), where it is bound to the demon instead. This adds a layer of mercy to what Slayers do, because it means that every staked vampire is a human soul reunited with its mind and able to move on properly.
So, to answer the main question, I consider Angel and Angelus to be separate entities (and separate characters, certainly), but only because Angelus lacks the portion of self that dominates Angel. They cannot be two distinct people, because a) without a soul, Angelus isn't a complete person at all and b) Angel contains everything that is Angelus within himself"”there are no components missing, just one added, which happens to be the one that makes all the difference. Angel is ruled by superego, Angelus by demon, and Liam by id.
I don't hold Angel responsible for anything Angelus did (nor do I hold Spike responsible for anything he did before he got his soul), but I would be very surprised (and a little appalled) if he were able to disassociate himself from it and not feel guilty. It may not be logical for him to blame himself, but guilt isn't a logical emotion, and what reason does he have to believe he's innocent of all of that when his own memories and nature tell him otherwise?
