I think my thoughts on this subject are well documented. As far as I'm concerned, the method of answering the question of Angel's identity, is starting with attempting to define what the soul is.
As I've said many times before, I believe in a dualist slant on the 'Verse. That is to say that there is a sense of consciousness that exists independently of the body.
Now some (who I call the Theory 1 supporters) adopt the concept that the soul equates to this consciousness. So this means the the soul and demon have two distinct consciousnesses and are effectively two separate individuals. This makes the task of holding Angel and Spike responsible for the things the demon did highly problematic.
My personal favourite however is Theory 2. This means that the individual has a 'spirit' which equates to their consciousness etc, but this is distinct from their soul, which is for want of a better term, a conscience.
So let us put this in simple terms
Theory 1
Liam= Human soul (spirit)+human physiology
Angelus= Demon Spirit and physiology- human soul.
Angel= Demon Spirit and physiology+ human soul (which is in control and somehow remembers everything the demon spirit did.
Theory 2
Liam= Human spirit+human physiology+human soul (conscience)
Angelus= Human spirit+demon physiology-human soul
Angel= Human spirit+demon physiology+human soul.
So under Theory 2, Angel is basically the same individual persisting through time under different 'conditions'. This the reason human Angel in IWRY, identifies more with being Angel than he does with being Liam. Basically because of the memories he's accrued through time. It's the same reason DarkWes wouldn't go back to being WatcherWes unless he lost his memories.
Different or the Same?
Under the second theory, Angel and Spike are the same Cartesian individuals they always were. It also makes more sense with their use of pronouns and their respective narratives. Angel talks about the things he's done and remembering it with a conscience. This only makes sense under theory 2. Essentially, Angelus was a complete bastard, so he was cursed with the ability to care about his past actions, which eventually led him to want to redeem himself.
Many people comment on the differences between Spike and Angel or more importantly, Angelus and Angel. My arguments has always been the reason why Angel and Spike react to their soul is more or less psychological. It's like, two soldiers going to war and one coming back with PTSD. We don't judge the soldier with PTSD, it's just that his brain reacted differently to the stimuli, than the other soldier's brain.
Also theory 2 makes more sense in context to these exchanges.
ANGEL
(stops, turns to Spike angrily)
Fair?! You asked for a soul. I didn't! It almost killed me. I spent a hundred years trying to come to terms with infinite remorse. You spent 3 weeks moaning in a basement, and then you were fine! What's fair about that?!
SPIKE
Well, that's just it. I am. And you know it. You had a soul forced on you as a curse. Make you suffer for all the horrible things you'd done. But me... I fought for my soul. Went through the demon trials. Almost did me in a dozen times over, but I kept fighting.
Angel only attacks his motivations, but not the actuality of what Spike said before.
When I have more time, I'll probably look into this more. The subjects, I'd like to look at are the sense of personal identity, guilt and responsibility and also the metaphysical mess that is Ats Season 4.