What are the advantages of using my own blockchain versus the Bitcoin blockchain? 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?
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.