What are the drawbacks of a paper based appointment system?

It can only be used by one person at once, so if one person is booking an appointment everyone else has to wait till the book is available to avoid double booking.

It has to be physically in one place to avoid double booking

With only one copy if that copy is destroyed or damaged ( which is quite easy with paper) then the entire appointments are lost. it is not easy to constantly make copies of an appointment system.

Because the appointment system can only be in one place it isn't possible for the person making the appointment and the person with whom the appointment is made to see it at the same time.

A book with appointments in it is not very secure and unless locked away can easily be read by anyone.

In short electronic systems are a lot more efficient,. and enable a lot more things to be done. e.g a doctor can book a person in for the future whilst the receptionist is booking other people in at the same time. More than one receptionist can do bookings at once etc etc.