No, Boxes 1a and 1b are not the same. Box 1a reports the "Ordinary Dividends" paid to you. Box 1b designates which part of the dividends in Box 1a are considered "Qualified Dividends". This is because the portion of your Dividends that are deemed "Qualified Dividends" are taxed at a lower tax rate.
Although it may appear that your Qualified Dividends are being taxed twice they are not. The amount of your "Qualified" Dividends are separated out so that they can be taxed at the preferential capital gains tax rate. When you have Qualified Dividends and Long-Term Capital Gains, your tax is calculated on the Qualified Dividends and Capital Gains Worksheet.
If you entered your Form 1099-Div Box 1 only, your tax liability would increase. After entering Box 1b, your tax liability would decrease.
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