Thanks for your question. A simple way to think about it is:
Public and private relates to the blockchain architecture based on ownership of the data infrastructure
Permissionless and permissioned relates to architecture based on read and write permissions granted to the participants
The common combination is public permissionless which is the most open form of blockchain where anyone can join, read and write and data is hosted publicly. This is where most cryptocurrency projects focus on.
The other end of the spectrum is private permissioned where only authorised participants can join and read. This is where most enterprise blockchain established by corporates focus on.