A parent or a legal guardian is the creator of the child account and can manage rights of the child. A parent can also manage goals and rewards. So a parent is an owner as well as the mentor of the child.
A person invited as mentor CANNOT manage child account. The only rights a mentor has are the ones given to him/her by the parent. So a parent is child account owner as well as a mentor. A mentor is just a mentor.
There is no difference. We mostly use the word 'mentor' because we believe a mentor could be a teacher or someone else (e.g. father, sister, mother, brother etc).
The 'My Classes' feature is meant for mentors who are mentoring many children. They would be using this feature to manage children. Most of the time mentors who are mentoring many children are teachers.