People use 'antisocial' colloquially as a synonym of 'asocial', but this is wrong, and asocial is the actual antonym.
Antisocial is no more an antonym of social than antisemitism is an antonym of Judaism.
You don't want to go describing other people as antisocial when asocial is what you mean, this thread illustrates that.
I joke with friends about this in fact, if someone says "ah I'm probably not going I've been feeling kind of antisocial" I'll ask them if they're planning to [redacted comment on America's violent culture in poor taste].
This is magnified by British use of antisocial, see ASBO, it's not worth trying to hold the line for two meanings when asocial is just sitting there being unambiguous.
Not in American English, which is why the Americans here are so confused about this whole thing. In American English, "antisocial" means "asocial" and the other meaning is obscure. And "asocial" just isn't in the lexicon as an alternative.
English is a muddle.