Public or Private Blockchains?

Bitcoin is the first iteration of a public decentralized Blockchain, controlled by no-one and cartel free.
What other options are there? To summarize, there are generally three categories of blockchain-like database applications:

• Public blockchains: a public blockchain is a blockchain that anyone in the world can read, anyone in the world can send transactions to and expect to see them included if they are valid, and anyone in the world can participate in the consensus process – the process for determining what blocks get added to the chain and what the current state is. As a substitute for centralized or quasi-centralized trust, public blockchains are secured by cryptoeconomics – the combination of economic incentives and cryptographic verification using mechanisms such as proof of work or proof of stake, following a general principle that the degree to which someone can have an influence in the consensus process is proportional to the quantity of economic resources that they can bring to bear. These blockchains are generally considered to be “fully decentralized”.
• Consortium blockchains: a consortium blockchain is a blockchain where the consensus process is controlled by a pre-selected set of nodes; for example, one might imagine a consortium of 15 financial institutions, each of which operates a node and of which 10 must sign every block in order for the block to be valid. The right to read the blockchain may be public, or restricted to the participants, and there are also hybrid routes such as the root hashes of the blocks being public together with an API that allows members of the public to make a limited number of queries and get back cryptographic proofs of some parts of the blockchain state. These blockchains may be considered “partially decentralized”.
• Fully private blockchains: a fully private blockchain is a blockchain where write permissions are kept centralized to one organization. Read permissions may be public or restricted to an arbitrary extent. Likely applications include database management, auditing, etc internal to a single company, and so public readability may not be necessary in many cases at all, though in other cases public auditability is desired.

Source: read full article:

On Public and Private Blockchains - ethereum blog 
by: Vitalik Buterin

Forbes 9 September 2015: Bitcoin’s Shared Ledger Technology: Money’s New Operating System
See also -IB Times UK, by Ian Allison, 8 September 2015:

Nick Szabo: If banks want benefits of blockchains they must go permissionless
Extract:

"Legendary cryptographer Nick Szabo says the only way to realise seamless, automated, and global financial integrity is if banks embrace permissionless blockchains, like the system Bitcoin runs on.

If banks were to embrace permissionless blockchains they could participate as well or better than the newcomers.

"But their bureaucracies are so heavily invested in the expertise and importance of local regulations and standards that it's extremely difficult for them to cut the Gordian knot and implement seamless global systems."
See also:

"There are many benefits to permissioned ledgers, but the cost of immutability provided by the most popular unpermissioned blockchain is so low, that it makes sense for even a permissioned ledger to utilize it as insurance.

By using the same SHA encryption technology that protects bitcoin balances, I would argue that either the central point (i.e. permission giver) and each of the players in a permissioned blockchain can create a full hash of their blockchain and record it in the Op_Return of a unpermissioned blockchain.  In fact I can see a future where permissioned ledger wallets/nodes vary their chatter to the decentralised network dependent on the level of risk they perceive in the permissioned ledger environment.

By treating a permissioned blockchain almost as a mutable coloured coin asset on a decentralised blockchain, this cheap insurance policy will act to strengthen the immutability of the permissioned ledger, along with the prestige and reputation of the global blockchain."

Read the full article: Permissioned versus un-permissioned Blockchain ledgers?

FT article:
Blockchain promises back-office revolution
"For now the market remains split between three core system approaches: private blockchains such as R3CEV and DAH, known as permissioned ledgers, which make use of trusted networks; open-ended blockchains such as Ethereum, which stay faithful to bitcoin’s original design but improve on the smartness of the ledger technology; and applications designed to take direct advantage of the prevailing bitcoin network."