They are definitely not used interchangeably since they have a very different meaning/purpose:

Private blockchains are private because of the genesis block they are using their blocks not matching any other block chains (you cannot link blocks from the main net to blocks of a private net with a different genesis blocks as the block header hashes won't match).

Note that any node can connect to a private block chain if it knows a bootstrap node address to sync and the network id. This node can perform any action on the private net; mine, make transactions, deploy contracts, etc.

Permissioned blockchains on the other hand offer an access control mechanism so that peers are allowed or rejected based on a control value (an address, a certificate, etc.).

They are definitely not used interchangeably since they have a very different meaning/purpose:

Private blockchains are private because of the genesis block they are using. Their blocks do not match with any other blockchains (you cannot link blocks from the main net to blocks of a private net with a different genesis blocks as the block header hashes won't match).

Note that any node can connect to a private blockchain if it knows a bootstrap node address to sync, the network id, and the genesis file. This node can perform any action on the private net; mine, make transactions, deploy contracts, etc.

Permissioned blockchains on the other hand offer an access control mechanism so that peers are allowed or rejected based on a control value (an address, a certificate, etc.).