A Snowball is another name for shaved ice. Shaved ice is ice that is shaved so fine that it has a light, fluffy, snow like consistency. This texture allows the ice to absorb the flavor better giving you a great tasting treat from top to bottom.
A Snow cone is finely crushed ice that is crunchy and consists of larger granules of ice. Due to the texture of the ice, it does not hold the flavor throughout as the snowball does and the flavor goes straight to the bottom giving it a bland taste. This is why you end up drinking the flavor from the bottom of the cup.