Public vs Private Blockchain – access based on user authentication (WHO are you)
Permission vs Permission-less Blockchain – rights based on user authorization (WHAT can you do).
Private Blockchain is a private network (like the intranet) that use the same genesis block and connect nodes between them by respecting a set of rules (rights, policies, digital certificates, VPN, etc). Any node can connect to a private Blockchain if it respects the set of rules, knows a bootstrap node address to sync, the network id, and the genesis file. This node can perform any action on the private net: validate blocks, make transactions, deploy contracts and interact with contracts.
Permission Blockchain - the participants in the network have the ability to restrict who can participate in the consensus mechanism of the Blockchain's network. Here is possible to have subsets of nodes that are validator, others that are transactions initiators, and others that are deploying or executing smart contracts. One or many participants in the network have the authority to manage different levels of access.
Examples:
Public and Permissioned-less Blockchain: Bitcoin – anyone can join the network, any node in the network can participate in the consensus algorithm (could validate transactions).
Public and Permissioned Blockchain: Ripple, Stellar or any POS Blockchain – anyone can join the network, only nodes that respect one rule are allowed to validate transactions (to be part of consensus algorithm). This rule is "to stake a minimum amount of tokens (coins)".
Private and Permissioned-less Blockchain: Private Ethereum – the "privacy" is ensured by allowing members to join the network if they have VPN credentials and digital certificates. Any node can participate in consensus algorithm.
Private and Permissioned Blockchain example: Hyperledger Fabric - the "privacy" is ensured by allowing members to join the network if they have access rights. The permission to validate transactions (in the consensus algorithm) is granted to certain predefined nodes.