When you initiate a chargeback, the merchant has the right to dispute the chargeback. If they can provide proof that the purchase actually took place, the chargeback will fail.
We don't know all the details of your situation, of course, but it appears from what you have said that the tax chain probably has documents that you signed agreeing to the charges. They prepared your return (even if they did a poor job), and so from their perspective, they have decided that they deserve to be paid. Whether or not they did a good job is a matter of opinion, of course; their position might be that they did it correctly, and the second business did it poorly.
The chargeback is a powerful tool, but it is not a magic button that makes a charge disappear. If the merchant can show that a sale did indeed take place and show that the proper amount was charged, the chargeback will fail. For a service, it isn't enough usually to simply state that you were unsatisfied; if you received the service at the agreed-upon price, the charge is valid.
A chargeback is sort of a nuclear option when it comes to getting a refund. There are negative ramifications and expenses every time a merchant gets a chargeback (even if they ultimately win), and so often they will be willing to work something out to avoid a chargeback. You should go to the merchant first, if you can, and ask for a refund before considering the chargeback option. If you file a chargeback without even giving them the opportunity to work it out with you, the merchant will usually want to fight back.