Normally, when a substance is heated, the temperature of the substance changes; however, during phase changes, the temperature of a substance stays constant. For example, if water is heated, the temperature of the water keeps rising till it reaches 100°C and then it does not rise any further until all the water turns into steam.
In your mouth, ice absorbs latent heat as well as the heat energy in order to attain your body temperature whereas Water at 0°C absorbs only heat energy to attain your body temperature. Thus water absorbs less heat energy from your body compared to ice. That's why ice feels colder.