Cryptocurrency is the payment that miners on the Blockchain system receive for their work. Basically, one cannot exist without the other as no one would take the time to mine information for long periods, for free.
The main distinction between public and private blockchain is related to who is allowed to participate in the network, execute the consensus protocol and maintain the shared ledger. A public blockchain network is completely open and anyone can join and participate in the network. The network typically has an incentivising system to encourage more participants to join.
A private blockchain network requires an invitation and must be validated by either the network starter or by a set of rules put in place by the network starter. Businesses who set up a private blockchain, will generally set up a permissioned network. This places restrictions on who is allowed to participate in the network, and only for certain transactions.