The advantage is the significant tax savings of donating assets that have grown (because you eliminate capital gains by donating assets that have appreciated). Instead of selling it off, paying taxes, and then donating the proceeds, you transfer the stock or whatever directly to the DAF, which then sells it, retains the entire value without paying taxes on it, and gives it to the charity of your choice. Local charities cannot usually receive donations of stocks or funds, but they can receive the check that your personal DAF sends out for you. This allows you to donate more money while simultaneously getting a larger tax deduction.
Also, as Nords pointed out, DAFs allow one to give anonymously as well as in other creative ways (think setting up matching donations, or providing an organization with a below-market-rate mortgage, etc).