Why is 0°C ice more effective at cooling a drink than 0°C water of the same mass?

When ice (or ice cold water) is added to a drink, the two substances move towards thermal equilibrium (the point where there is no net energy transfer between the two substances, and they have the same temperature). For this to happen, the ice must warm up, and the drink must cool down. Remember that temperature is a "measure of the average kinetic energy of the particles in a substance"... but kinetic energy is not always the only form of energy present in a substance! Ice cold water, in liquid form, contains latent energy (the energy that was required by the ice to change state into water), so already has more energy than the same amount of ice at the same temperature. Therefore, ice cold water is not as good at cooling a drink as solid ice is, because ice starts with less internal energy, so the point of thermal equilibrium is reached more quickly and to a colder temperature.