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Let's say I have a project that takes advantage of blockchain technology. Now I see a number of banks dropping the Bitcoin blockchain for their own. I'm sure that has something to do with secrecy, but, assuming I'm just a normal entrepreneur and not some massive financial institution, what would be the reasons for me to create my own blockchain vs. the Bitcoin blockchain? What are their respective (dis)advantages?
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The Bitcoin blockchain is maintained and secured by people throughout the world: miners. They take care of the security and validation of the decentralized consensus rules. The incentive for those people is the bitcoin reward.
Your own blockchain will need such people to secure your blockchain. How are you going to incentivize them to mine for your blockchain? Is it going to offer some service in return and/or a reward in an alt coin (like the majority of altcoin blockchains) ?
To answer your question, I see no reason to create your own blockchain unless you have a use case that noone have thought until now. There are several hundred different blockchains already. The vast majority are bitcoin clones that offer nothing new and have much weaker security than bitcoin. Others, like Dash offer nice additions and just a few, like Ethereum, offer unique functionality on their blockchains.
Even for massive financial institutions I would argue that they will not gain much if they build a blockchain in isolation. One application I see is several such institutions joining to support one blockchain. So they share the resources of securing the blockchain, they have part of transactions public for auditing but still control what is private and what not (permissioned blockchains).
We would need to know the requirements of the hypothetical project to be more specific. However, I do recommend you to read this and this, since it goes into detail on requirements for financial institutions and governments and permissioned blockchains.
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As Bitcoin blockchain positive sides are covered in the answer above here are some negative sides
You cannot change the rules - block time is 10 minutes, there is so much text data you can add by default, etc. This makes Bitcoin transactions non-trivial for more advanced transactions.
Even Bitcoin community itself seems to be unable to change its own rules - the current debate about how to make block sizes larger has shown weaknesses in the community and inability to have a resolution.
Bitcoin is de facto currency of darknet markets and drug trade, so it implies a lot of crime and money laundering. This negative image may harm public relations of any Bitcoin blockchain based project.
However Bitcoin is not only public blockchain out there. If you are building anything custom you should see out at least Ethereum, Stellar, NXT and Bitshares which have more advanced features built into their respective protocols.
Please also see that the common argument "secured by miners" may not really apply to your use case; for example consensus based Stellar works with low number of participants even if there are no miners. The future Ethereum protocol (Casper) is migrating to proof-of-stake instead of proof-of-work making miners obsolete. In fact miners are rarely present in any of blockchain technologies currently being developed.
If you want to see projects that innovate on the top of Bitcoin blockchain see Counterparty. They are facing Bitcoin constraints like the requirement of using two blocks for certain transactions, making instant-like transfers already infeasible.
