Ha, this sounds so much like the typical uber-empath, sociopath relationship. I have loads of friends like you. If this really is the situation (e.g., he is a sociopath), he might be very startled by and wary of your apparent willingness to accept him for who he is without judgment. It is possible, as you suggest, that he is a sociopath and knows that he is different in some way, but doesn't know that he is a sociopath, per se. He may be like many of my sociopath readers, questioning why he is the way he is, while actively avoiding being saddled with the term sociopath (for reasons that are probably obvious to you, particularly after reading The Sociopath Next Door). He may be particularly wary of you given that you "confronted" him about his behavior during a conversation in which you were attempting to break up with him. He probably reasonably interpreted this to be disapproval.
I wouldn't necessarily say that sociopaths do not have the ability to love, although theirs is not a typical form of love. I wrote about it here. It sounds like he is genuinely fond of you, if he put forth extra effort to win you back. It also sounds like he also respects you, if he has toned down the lying. He is probably charmed that you were cognizant enough to see through him, and intrigued that you appear to know who he is but are still interested in him. Everyone wants to be known and appreciated for who they really are -- sociopaths are no different. Sociopaths tend to take a lot of pride in their work, for good reason, so I am sure he would enjoy the thought of having an audience. On the other hand, you never know who you can trust. I live in daily fear of being outed. I enjoy writing the blog and am glad that it is helpful, but I frequently think that it will eventually be my doom, so I understand that sort of apprehension. And even if you were willing to listen without being judgmental, there is no guarantee that you would be able to handle the truth, wouldn't freak out and disclose everything. He may also have something else to hide. Sociopaths can do bad things. But if he did admit it to you, what would you do? Be a willing accomplice in his life, no matter how sick it turns out to be? Watch the film Let the Right One In for an example of that sort of dedication and to what it can lead.
The way you describe him, though, I can see why you are fascinated.