A Public Blockchain is a peer-to-peer network that is accessible by everyone. To clarify, a blockchain network is a distributed ledger of transactions. The distributed nature of these ledgers allows for transactions to be verified by everyone on the network, which makes these networks more transparent and secure against hacking. On a public blockchain, anyone is allowed to join the network and start verifying transactions.
A Private Blockchain is a peer-to-peer network that is only accessible by invite. This means that someone currently using the network must send you an invite to be able to join the network. This ensures that only trustworthy parties are allowed to verify transactions.
THE MAJOR DIFFERENCE
Ultimately, the difference between a public and private blockchain network boils down to who is allowed to participate. This can be crucial based on the scale of the project implementing the blockchain. Essentially, the trade-off for using a private blockchain is losing access to the masses for the sake of a more trustworthy network. Whereas cryptocurrencies running on a public blockchain have much higher market caps, but this comes at the cost of requiring a large number of computational resources, issues with scaling and more malicious parties on the network.
All in all, within the confines of each network, the underlying blockchain technology functions the same way, including the security and decentralization features. Depending on the scale and sensitivity of the project, there are benefits and drawbacks to both private and public blockchains.