>Without the incentives, distributed computing power, and public participation in a blockchain like Bitcoin, what does a private blockchain actually do?

Proof of Work, public networks make most sense for low trust scenarios where you wish to move actual money with people you don't know or trust at all. There are other scenarios however.

A few examples:

Scenario 1: Large Banks eliminating middlemen and making settlement for more cost effective

Quoting from another comment I made:

> It takes some financial assets 3-5 days to settle after the initial transaction goes through, weaving its way through many database systems (some many decades old) and middlemen who each take a cut, e.g. clearing houses. Any established industry has lots of cruft that builds up.

> For this reason Santander estimates that banks can save $15-20bn per year collectively by using blockchain technologies to make their infrastructure more efficient[1][2]

[1] http://www.coindesk.com/santander-blockchain-tech-can-save-b...

[2] http://santanderinnoventures.com/fintech2/

From another comment:

> Since with a private banking settlement blockchain you know in advance who the players are (individually-invited banks) you can give them a percentage of the network and have each node verify the transactions across the network. You could even give nodes to regulators or consumer advocacy groups to achieve regulatory compliance or consumer oversight if required.

> As for cost, you could technically run a PoS node with a raspberry pi, a large USB drive, a wifi connection, powered by a solar panel and an old car battery on the roof of a building. Proof of stake only needs minimal computing power and a persistent internet connection.

Saving billions in costs at a time when banks are struggling to increase revenues is a powerful incentive to consider blockchains. You don't need a public blockchain because you know in advance who the players are.

Scenario 2: Streamlining KYC data, therefore improving data quality and massively reducing costs

I sat at a presentation where a startup outlined a product roadmap being offered to large financial institutions such as banks or insurance companies. A big problem is KYC data gathering- often the customer data goes stale, therefore data quality is poor, and it's expensive, costing about €200 per account to process.

In the proposed product, a private blockchain is implemented to gather and manage customer data, hosted collectively by the institutions that are interested in cost savings and efficiency improvements. The customer decides when, with who and how their data is managed - a big improvement over today. If you update your address for Bank A - then your insurance company, your second bank, etc all get your updated, verified address. If your car insurance is a 12 month contract, you can terminate access to your personal data after the contract is up. If you decide to get car insurance with a second company, you grant permission to that company, who then doesn't need to see any form of ID because your identity is cryptographically verified (technically they don't even need to know your name!)

The cost to gain access to the customer data is borne by the company getting access, a fee which they're happy to pay because it's vastly cheaper to pay that rather than gathering and securing customer data themselves, paying a company to verify your ID, and so on.

These are the early days of blockchain technology. There's lots of potential advantages of a distributed database over a SQL database, and we're only starting to figure out what those are, and what scenarios where is doesn't make sense.