Public Sector bank means any Government Sector 
Bank/Institute that goes public... means that issues it 
share to general public.. It also has a greater share of 
government(more than 50%) so that the main motto of social 
welfare other than Maximising Profit remains.

Where as Private Sector Banks are those Banks where the 
management is controlled by Private individuals and 
Government does not have any say in the management of these 
banks.Maximising profit is the basic motto.