It should be pretty clear that the private party controls the network and decide the protocol rules. Not everyone can participate in the private blockchain to verify the transactions.
The term "public and private blockchain" often considered different from "open and closed blockchain" or "permissioned and permissionless blockchain" but in my opinion, they are pretty much the same.

These blockchains have a very specific use case, unlike the open/permissionless/public one. Maybe consortiums of banks or companies will use it to smooth their operations.