Let me clarify this good question by saying that in either case, you are taking the dividends into income and then making a charitable contribution for exactly the same amount, so there really is no difference between the two scenarios. If you gave all of your shares of the mutual fund to the charitable organization and a dividend was paid, you would not be taking the dividend into income and would not be getting the deduction unless you were using some advanced estate planning techniques that are beyond the scope of this question. In addition, I assumed that you will transfer to the charitable organization just the shares that were purchased with the dividend when it "rolled over into the fund" (this is known as a Dividend Reinvestment Plan or a "DRIP" plan).

In both cases, it is a tax neutral event if you are itemizing your deductions because in both cases, you are still taking the mutual fund dividends into income and then taking an equivalent tax deduction for them 
when you give the dividends (or give the shares purchased with the dividend) to the charity. As you may be aware, in order to take a charitable deduction, you must itemize your deductions. However, if you contributed all of the shares of the mutual fund to the charitable organization, some special rules apply as outlined below.

If the mutual fund has been held for ONE YEAR OR LESS and has appreciated in value during that time (the share price has gone up), you will receive a charitable deduction for only the ADJUSTED COST BASIS of your mutual fund. The adjusted cost basis is equal to your initial investment in the fund plus any dividends that have been "rolled over into the fund."

If the mutual fund has been held for MORE than one year plus a day and has appreciated in value during that time, you will receive a charitable deduction for the FAIR MARKET VALUE of the mutual fund at the time of the contribution. This way, the charitable contribution INCLUDES the appreciation in the mutual fund, which means a much larger charitable contribution for you.

One final thing to be aware of (and you can see why some folks want to abolish the current tax code) is the income limits on charitable deductions. For "public" charities (churches, schools, hospitals and the 
like), your total charitable deduction is limited to 50% of your adjusted gross income (the last number on the first page of your tax return) if you are giving mutual fund shares that have been held for one year or less. If you are giving property that has been held for more than one year plus a day, your total charitable deduction is limited to 30% of your adjusted gross income.