In the opening chapters of Animal Farm, Snowball and Napoleon share the common belief in overthrowing Mr Jones and taking control of Manor Farm for the good of the animals. Together, for example, the pigs develop a social system called Animalism which emphasises the equality and freedom of animals and frees them from the exploitation of Man.
But it is Snowball and Napoleon's differences which are most prominent throughout the rest of the novel. We see lots of examples of this after the revolution takes place and the humans are driven off Manor Farm. On the subject of clothes, for example, Snowball throws some ribbons on to the fire because he believes that they are the "mark of a human being." He states that animals should never wear clothes and should always be naked. This contrasts sharply with Napoleon, who wears human clothes later in the novel. For Napoleon, then, clothes are a symbol of his power and dominance and he is proud to wear them.
Similarly, after the revolution, Snowball believes that the animals should continue to work hard on the farm to ensure its smooth running. For him, it is a "point of honour" to work harder and more efficiently than the humans. In contrast, Napoleon does not value work in the same way and he uses this opportunity to steal the milk.
Snowball also believes in forming committees to improve the lives of the animals. While many of these efforts are failures, his reading and writing initiative is a "great success." But Napoleon believes that the "education of the young" is far more important. He does not participate in any of these committees and he instead takes Jessie and Bluebell's puppies on the (false) pretence that he is educating them privately. In reality, Napoleon is training the puppies to become his own personal guard dogs and does not value education at all.
Finally, we see the differences in Napoleon and Snowball's beliefs at their strongest when it comes to the building of the windmill in Chapter Five. Snowball believes in the value of the windmill because of its labour-saving potential. He also believes that it can greatly enhance the comfort of the animals by giving them heating and lighting in the barn. Napoleon, however, feels very differently about the windmill. For him, the windmill is a waste of time which detracts from the animals' most important mission: boosting food production to prevent starvation. But when Napoleon loses the windmill debate, he reacts with unprecedented cruelty by setting his dogs on Snowball. What this incident really shows is that Snowball and Napoleon differ on the most fundamental level: Snowball is concerned with the welfare and equality of all animals, while Napoleon is only interested in the pursuit of personal power.