CFI wants to know the status of a students training.

With a paper PTR logbook, the CFI would likely call the flight school administrator, who would probably reply "I'll have to check and get back to you".  The administrator would then have to go find the paper logbook, look over it, total up the ground school time, check the totals for the flight times and then call the CFI back with the information.

With an online PTR the CFI could login to the system themselves and within a couple of clicks have the complete answer.

An instructor gets a call at home from their school and is asked to take a training flight later that day covering for another instructor who is not available.  As the new instructor has not flown with the student before, they need to know which lessons the student has covered and what they are currently working on.

Using an online PTR the instructor can login to the system from home, or with their cell phone, and review the students PTR and be prepared for the lesson before they arrive at the school.

On the way home, after a flight with a student, an instructor thinks of a comment to put in the students PTR.  Using a standard paper PTR the instructor would either have to call the school to have someone enter the comment in the PTR, or write it down somewhere, to hopefully recall it for the next flight.  Using an online PTR, the instructor can login to the system, even from their phone on the way home, and directly enter the comment in the students PTR.