I think apathy of ignorance is the worst thing. If you don't know you don't know then no harm no foul. If you know you don't know and you have no drive to learn then I have a issue with this person.
Apathy is the tougher nut, as you have to determine the root cause of the apathetic behavior, correct/cure it, then engage the ignorance once the person again want to learn.
Apathy is worse, caused by a pain usually received by the hand or actions of another. This may take several years to overcome and only if the person is willing to work on it in the first place. Ignorance is just the lacking of education one's self in any matter or aspect of life's work and is a easy fix for most of society. If there is a person who is afflicted with both then the apathetic nature must in my opinion be resolved first as it being the worse of the two.