What does the "Timesheet Administrator" system role allow?

The Timesheet Administrator role allows the User to see everyone's timesheet. 

With this enabled, a Timesheet Administrator can logon to the timesheets and see the timesheet user selector on the top right had side of the timesheet. 

It is important to distinguish between a Timesheet Administrator and a Timesheet Approver. A Timesheet Approver differs as this is someone that approves timesheets they have been assigned to on an individual user basis on the User profile-Employment Details tab. 

This gives them the ability to approve only that User's timesheet. This is only applicable if the approvals setting is enabled on the configuration. 

They can therefore approve/reject timesheets submitted by users on a weekly / monthly (depending on the configuration) basis via the My Timesheet approvals widget. This Timesheet Approver does not need to have the Timesheet Administrator role enabled in order to do that task.