Ice at 0 deg Centigrade and water at 0 deg centigrade...
Ice needs heat to melt into water. This is called latent heat of fusion. This is some hundreds of calories..
So water at 0 deg C, has more heat in it than ice at 0 deg C.
So ice absorbs more heat readily, and becomes water, This process takes in more heat. So it cools better. So when ice becomes water, it already cools the given substance a lot. So ice will keep the given substances cool for a longer time.
Water at 0 deg. C, absorbs heat to become hotter. So water becomes hotter much quicker than ice at 0 deg C.