Your "My Notes" section is private just to you. Neither students nor other tutors can see these notes. You can add to "My Notes" in-chat (or afterwards by clicking on a student's profile from your History page).
The "Public Profile" is viewable and editable by both tutors (in-chat) and the student. Tutors are able to edit the Public Profile as the student's goals and bio changes, making the Public Profile a single point of updated, relevant information on the student.