It is a real shame that you can only add one copy of the Links block, but since you can add multiple HTML blocks, it is still easy to do what you want.
Since you didn't mention what roles and what version of Moodle, I am going to guess at student and a role you created called tutor, and that you want to do this on Moodle 3.
Add your two blocks, let's call them Student Links and Tutor Links. Go edit the permissions for Students Links and remove View block permissions for everyone but Student, Non-editing teacher, and Teacher. Go edit the permissions for Tutor Links and remove View block permissions for everyone but Tutor, Non-editing teacher, and Teacher.
Once you've done both permission changes, student can only see their HTML block and Tutors can only see theirs. Teachers can still see both and edit both.