Public blockchains can be used by anyone and are perfect for public services. Private blockchains require permission to use and are great for businesses solutions.
Public chains enable anyone to run nodes and to add and explore data. They implement cryptographic (crypto) coins for several functions. The first is project funding, resembling company shares. The second is spam prevention, imposing transaction fees to prevent the system from flooding with data. The third is public node compensation for electricity bills, server costs and locked stake.
Private (permissioned) chains require user permission to run nodes and to add and explore data. Nodes are run by consortium members who have an incentive to keep the system operational. System users can be added and removed. There is no need for crypto coins as spam prevention and node rewards are not required.
Public vs private
Public chains are a perfect choice for public services such as digital currencies, but it is difficult to store large amounts of data on, application development requires rigorous security testing and the technology has not fully matured yet. Private chains solve these problems making them a great choice for business solutions, but are less suited for storing large amounts of value owned by the public.