That's the difference between a sociopath and a psychopath.
A sociopath feels emotions but can't understand them, leading to them being outcasts who can't understand people.
A psychopath understands emotions but do not feel them, leading them to be charismatic people who can manipulate others.

Arthur wishes he was patrick, patrick fears no keeping up the illusion will turn him into Arthur.

Arthur Fleck was more a product of abuse and disrespect, sure there's something wrong with him and he'll likely never function properly in society, but if you were to restrain him to an institution and keep him properly taken care of and monitored before the events of the film unfolded, I don't think he'd be a problem patient that starts fights frequently. Bateman is the absence of soul and spirit, having only an animalistic drive for a status akin to Alpha Male, whereas Fleck is more of a tortured soul, starting off as simply mentally ill and in need of treatment before the events of the film permanently warp him into the irredeemable monster known as The Joker. Bateman does not want to be accepted, only feared, Fleck initially does not want to be feared, only accepted until the events of the film unfold and he gains acceptance in the form of fear. Bateman has no aspirations besides a vague and shallow definition of success, whereas Fleck actually had hobbies and things enjoyed and enjoyed people's company outside of just what they could give him (even if it was just a hallucination), while Bateman merely wanted people for either financial or social gain or base pleasure, whether in a general sense or to satisfy one of his murderous urges.

Actually, now that I've said all that, I really want to see a film where these two fight to the death just for the philosophical arguments that it could spawn.