There's certainly a difference between when the average person says they don't feel, and when a psychopath does, but it certainly lies in the realm of possibility.
The definition for psychopathy changes within various countries and even the industry addressing the issue.
The US conflates what have clear distinctions where I am: sociopathy and psychopathy.
The definition Wikipedia leans towards would be called sociopathy here.
APD, and several other disorders, get classified as psychopathy here.
The definition I gave is simply the definition I was given post-diagnosis.
However, my point that a lack of emotion is possible, seems to stand regardless of the name you give to the disorder.
It's in my experience that psychopathy (or whatever we'd like to call it) is a consciousness state, and there are some people who can deliberately shift into it by choice. These are usually folks who experiment with hacking their consciousness, such as psychonauts or meditators, so I'm not necessarily talking about people who are born into it, or conditioned by circumstance into it.
The most interesting thing about consciously shifting into that state is that, once you are there, you feel zero motivation to shift out of it, even if you have the knowledge and skill to do so.
And it is not so much that there is a lack of emotion, so much as the lack of suffering when it comes to experiencing things. At that point, any morality is driven by choice, rather than by pathos.
An article: http://well.blogs.nytimes.com/2016/03/18/an-experimental-aut...
says: no psychiatric or psychological organization has sanctioned a diagnosis titled "psychopathy"
The words "psychopath" and "sociopath" are very general terms for mental illness.