A blockchain is a special kind of database. A very slow, expensive, inefficient, limited, poorly scaling database. But it comes with one HUGE advantage: decentralization. You don't need to trust or rely on anyone to have a reliable, immutable, trustworthy, secure database. No single point of failure, and completely trustless. This is a very unique, powerful, significant, never seen before property of databases.
For a few years now, people have come up with the notion of a "private blockchain". This is utter nonsense. If you take away the independence, decentralization and immutability, there is exactly zero reason to prefer a blockchain over a regular database. A private blockchain is like email on paper.
Now, people will say that a chained block structure, or clever cryptography, or usage of hashes or whatever makes their database secure, immutable, trustless, et cetera. It doesn't.
With a normal database, there is someone who owns or controls or facilitates the database. They can change its history. They can modify its content. They can bend the rules and add data that doesn't meet the criteria. They can take it offline. Or they can be forced to do so by a government, by malicious or incompetent employees, or a blackmailing competitor, or hackers, just to name a few examples. With a true blockchain, all these scenarios are fundamentally impossible.
You may use a distributed cloud structure with redundancy to mitigate the risk of server failure, but there's still a single point of control.
You could use hashes and checksums for integrity checks, but not in any way does that prevent someone from changing history and changing the hashes accordingly.
You might use some clever chained block data structure, but that is of no help at all if suddenly the blocks or links don't match.
All the advantages of a true blockchain, which prevent all the above scenarios and risks, do not apply in a private blockchain situation.
A so called "private blockchain" basically lacks ALL the properties that make a blockchain a blockchain.
Appendix - here's a nice flowchart:
