I think this depends on the country where you are studying. In the U.K private classes or tuition would mean you are receiving lessons that you pay for and either receive at your home address or at some mutually agreed on place. Or the same arrangement that your parents are paying for. Tutoring classes would be the same whether Public (state or charity funded) but we would usually just say tutoring or "classes" or "lessons" or "doing/taking classes with a tutor /private tutor" My tutor

A tutorial class is not necessarily private. I think it would be fair to call almost any private class a tutorial class. Private usually means for one person, but sometimes, might just be literal, not open to the public and for more than one person. In that case, I guess it could be private and not tutorial.