No one can explain what happened without being able to see all of the relevant documents: your federal and New York tax returns, your W-2s, and the notice from the state. We can't see your tax returns, and we can't get into that kind of depth here in the online community. (Don't post any documents here, and keep in mind that this is a public web site.)
Even though the state calls it a computation error, that doesn't necessarily mean that it was TurboTax that miscalculated. It's possible that you entered something incorrectly or answered a question wrong. It's also possible that the notice from the state is wrong. But again, no one can tell without seeing all the documents.
You could try calling TurboTax support and see if they will go through it with you.
How do I contact TurboTax?