I saw this sentence in a book of accounting which put me in doubt that there is some difference between the two, the sentence is "The balance of the Owner's capital account in the ledger will now be the same as the amount of Owner's equity appearing in the balance sheet."

In a sole proprietorship, you will have the Owner's Capital account and the Owner's Drawing account as part of the Owner's Equity. The Drawing account is temporary and is closed into the Capital account as part of closing entries. I'm guessing that statement you quote is referring to this. In practice, we use the terms owner's capital and owner's equity interchangeably.