The manual calculation assumes that all income is being taxed at the same rate which is not always the case. There are times when you need to use other methods to calculate your tax due - one reason being tax rates can be different depending on the source of income.
This is especially true when you have dividends or other capital gains as sources of income. If any of your dividends were qualified dividends, then capital gains tax rates would apply to those dividends. Capital gains tax rates are often lower than regular tax rates. In these cases, TurboTax uses the Qualified Dividends and Capital Gains Tax Rate worksheet to calculate your tax due (per the IRS). If you print the pdf of your return with all forms, you should be able to look at this worksheet.