One of the things that Piggy fears is being blinded. If you look at his reaction to the theft of his glasses and his incredibly childish and overblown fear while they travel to and confront the boys at the "fort," you see that he has a deep seated fear of losing his ability to see. Even when the boys first grab his specs in order to light the fire you see the incredible terror induced in him when he is without his glasses.
Ralph fears that he is too weak to really be a leader. Even at the outset when it appears that he has been elected chief, he is afraid to take the lead and to make decisions and act on the feelings he has.
Piggy fears that Ralph will give up being chief and allow Jack to tyrannically rule freely over the boys. Piggy knows that without Ralph, he will not be able to defend himself against Jack and his savages. Piggy also fears total anarchy among the boys. He is the biggest supporter of civility on the island and cannot survive in a society without rules, laws, and structure.
Ralph's main fear is never getting off of the island. Unlike Jack, Ralph does not want to spend the rest of his life on the island fighting to survive and wants to go home. Ralph also fears making poor decisions as the boys' leader. He understands that he is not as intelligent as Piggy and has no idea how to be chief, yet he wants to make the right decisions. Ralph struggles to effectively lead the boys and fears letting the entire group down.
Ralph attempts to use reason to dispel the littluns' fears. But Ralph is "facing something ungraspable," and he can only react in frustration with a blanket denial: "But I tell you there isn't a beast!" To the littluns, this assertion is utterly inadequate. Their fears are not rational. Ralph somehow understands this, but can't quite articulare it; nor can he find words to allay the children's terror.
This is symbolic of the novel's ongoing conflict between the rational and the irrational, between the cerebral and the instinctual.