The conflict between the pilot and the prince is certainly a bit complex. The conflict stems from the fact that the pilot is focused on his survival, trying to fix his engine so that he can escape the Sahara, while the prince has already realized that there are other things that are more important in this world than one's own life, especially love and the life of others. The pilot is actually not very far away from achieving this revelation. In fact, we see from his story concerning his drawings that as a boy he had already learned to value the things unseen. The prince is merely helping the pilot reiterate that point or take it one step further. Since this conflict involves two characters' thoughts, we can refer to this conflict as character vs. character. Or, if we wish to interpret the prince as a mirage or hallucination the pilot is having while in the desert, we could even call this conflict character vs. self.
We see the conflict of what's important vs. less important in several places. The first is when the prince appears at dawn out of nowhere asking for a drawing of a sheep. The pilot becomes annoyed because the prince keeps rejecting each drawing and the pilot is "in a hurry to start taking [his] engine apart" (Ch. 1). Through this scene we see that the pilot is more concerned about the future, his future escape out of the desert, then about the present feelings of others. However, we soon understand that the sheep is symbolic of the prince's own spiritual awakening. Like a sheep, the prince has strayed far from the things that are truly important to him, his home and his flower, and is now returning to the fold. Therefore, the sheep truly is a matter of grave consequence.
A second instance in which we see the conflict of important matters vs. unimportant matters portrayed is when we first learn about the prince's flower. The prince is very concerned that his new sheep might eat his flower, and the pilot can't see how discussing whether or not a drawing of a sheep will eat a flower can be more important than fixing his engine, as we see in his line addressed to the prince, "Don't you see--I am very busy with matters of consequence!" (Ch. 7). However, the prince very wisely retorts that a sheep eating a beloved flower that is universally unique is a matter of grave consequence, as we see in his lines:
And if I know ... one flower which is unique in the world, which grows nowhere but on my planet, but which one little sheep can destroy in a single bit some morning, without even noticing what he is doing--Oh! You think that is not important! (Ch. 7)
In other words, the prince is rightly arguing that it is his love for his flower, his love for another rather than for himself, that is of the greatest consequence, rather than escaping out of the desert to protect only one's own life.