Added energy will break the bonds holding the ice together instead of raising the temperature.
In ice, the bonds that hold one water molecule to another have to be broken before the temperature of the "water" can go up. This means that it has the ability to absorb some heat (cool something else) without raising the temperature of the "water". That is what another poster called latent heat of fusion.
If the ice/water mixture is nicely mixed, the temperature will not rise until all the ice is melted. Once melting has completed, the temperature of the water will rise as heat is absorbed from something else.
So in the end, the ice at 0 celsius can cool more effectively than the liquid water at 0 celsius because the heat that it absorbs at the beginning won't raise the temperature because that energy is used to do the melting. Once the ice is melted, it is just like the liquid water in your original question.