'Apathetic' describes someone who lacks the will to take action or who is not interested in doing so. They may know there are things they need to do, but they just can't be bothered. - He's so apathetic. He knows he should work hard to pass his school exams, then get a good job to help his family out. But he just lies in bed most of the day, looking at his mobile phone or sleeping. 'Unsympathetic' describes someone who cannot (or refuses to) relate to the feelings of others who are in difficulty, in pain or just unhappy about something. - I don't bother telling her about my problems any more. She just doesn't want to listen and acts like she couldn't care about what I have to live through.

I believe apathetic is more general. As in a person who generally shows no feelings and is indifferent. Whereas unsympathetic is more lacking sympathy and not general indifference.

They are synonymous. There's no difference in the meaning of the words.