The softness of ice cream arises out of the nature of the ingredients and the way they are combined. Two essential ingredients of ice-cream are fat in the form of cream and air bubbles introduced by stirring. As the water in the ingredients starts freezing, the suspended fat globules prevent the whole content from freezing as a single mass. Tiny air bubbles trapped in the mixture due to continuous stirring of the ingredients also prevent the ice crystals from growing as the temperature is reduced further. This prevents the ice cream from becoming hard and the frozen mass remains soft.
In the case of ice, the absence of fat and air bubbles leads to the tiny ice crystals growing into large chunks of hard ice.