How are Custom Sharing Rules different from the Public mode? 

The Public mode allows everybody to access your records. The idea behind advanced or custom sharing rules is to provide selective access to groups and roles, and even subordinates. 

For instance, a Sales Manager is at a higher level than a Sales Rep in the role hierarchy. When sharing rules are set to Private, a Sales Manager can view a Sales Rep’s records, but the other way around is not possible. 
Now, the Sales Manager wants to give access to his records to a Sales Rep. Setting sharing rules to Public makes the records visible to all the users. Setting up custom sharing rules gives the Sales Manager the flexibility to grant exclusive access to the Sales Rep.