A Performing Right is simply the right (ie 'permission') to perform a work (song, instrumental, etc.) in Public (such as a bar, club, restaurant, etc.). In this context Performance also includes playing a recorded version of a work, such as from a CD. The payments received from these uses are paid to the Writer & Publisher of the work.
Performers Rights are the rights of a Performer to receive payment for the use of their Performance in Public. This would include a song being played on the radio or the Performer doing a gig in a club.