A snow cone is a dessert made of granular shaved ice topped with flavored syrup.
Italian ice is smoother than a snow cone, because its ingredients are mixed while freezing. It consists of fruit juice or other flavors, water, and sugar.
Although sorbet is made with basically the same ingredients as Italian ice, it is smoother still. Alcohol may be added to sorbet, which prevents it from freezing completely, and augments its smooth texture. Sorbet is usually considered to be of a higher-quality, and more expensive than Italian ice.
Sherbet is essentially sorbet to which milk, egg white, or gelatin have been added. Sherbet contains between 1% and 2% milkfat. (By comparison, ice cream has at least 10% milkfat.) ‘Sherbert’ is just an alternative spelling and pronunciation of ‘sherbet’.