Assignment

The top one is not a normal Student view. it is the normal Teacher view because only Teachers can see the Grade and View all submissions button. In fact, the "Grade"/"View all submissions" button and the "Add assignment" should never show up for the same person.

The first thing to verify is that the Student and Teacher role permissions are correct in Site admin > Users > Permissions > Capability overview.

The following two capabilities should be set to 'Allow' for the Student and 'Not set' for Teacher (and all other standard roles):

1) Submit assignment
mod/assign:submit

2) View own submission summary
mod/assign:viewownsubmissionsummary

The following two should be set to 'Allow' for Teacher (and Manager and Non-editing teacher) and 'Not set' for Student (and all other standard roles)::

3) View grades
mod/assign:viewgrades

4) Grade assignment
mod/assign:grade

The following should be set to 'Allow' for Teacher (and Manager) but for Non-editing teacher and Student (and all other standard roles)::

5) Review grades
mod/assign:reviewgrades

If those are correct, then look for the issue where a user has more than one role in the course, or is inheriting a role from above, such as from a Category or the System level roles.

Also verify that your Default role for all users is Authenticated user in Site admin > User > Permissions > User policies, just in case that may be causing problems.