The sole distinction between public and private blockchain is related to who is allowed to participate in the network, execute the consensus protocol and maintain the shared ledger. A public blockchain network is completely open and anyone can join and participate in the network. The network typically has an incentivizing mechanism to encourage more participants to join the network. Bitcoin is one of the largest public blockchain networks in production today.

A private blockchain network requires an invitation and must be validated by either the network starter or by a set of rules put in place by the network starter. Businesses who set up a private blockchain, will generally set up a permissioned network. This places restrictions on who is allowed to participate in the network, and only in certain transactions. Participants need to obtain an invitation or permission to join. The access control mechanism could vary: existing participants could decide future entrants; a regulatory authority could issue licenses for participation; or a consortium could make the decisions instead. Once an entity has joined the network, it will play a role in maintaining the blockchain in a decentralized manner…

This type of permissioned blockchain model offers the ability to leverage more than 30 years of technical literature to realize significant benefits. Digital identity in particular, is fundamental for most industry use cases, be it handling supply chain challenges, disrupting the financial industry, or facilitating security-rich patient/provider data exchanges in healthcare. Only the entities participating in a particular transaction will have knowledge and access to it — other entities will have no access to it.

Source: The difference between public and private blockchain – Blockchain Unleashed: IBM Blockchain Blog