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][1]). 

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.). 


  [1]: https://blog.ethereum.org/2015/11/15/merkling-in-ethereum/