First, if you are under 59 1/2 years of age, there is an early distribution penalty. Since it is also 10%, I can't tell if this applies to you or not. This penalty is in addition to any income tax that you owe on the distribution.
Since the financial institution probably has no idea of your personal tion, the 10% is usually just a guess for withholding - that's why many use 20% instead.
Second, the tax collected on the distribution does not directly pay your tax on the distribution. Instead, it is held by the IRS as a credit against the tax calculated for your return. So the distribution will be taxed independently of any withholding on your return, and at the end of the 1040, the amount of withholding from all sources (1099-R, SSA-1099, W-2, etc) will be applied to the tax.
