A Separate Peace is a lasting favorite because it feels familiar: we've all had friendships with conflicts. We all feel some guilt about certain past choices. We all wish we could be better people than we are.
Gene envies Finny because Finny is such a great athlete, so popular, so intelligent. But Gene isn't completely honest with himself, and that combination--envy and self-deception--leads Gene to do things he can't explain and will regret.
The conflict between them keeps growing until it spins out of control, and as you read you feel more and more like shouting at Gene to wake up and look at himself. That feeling--caring about what characters are doing--is a sure sign of a good novel.
There are also some symbols, allusions, metaphors, and other stuff that English teachers like to make you think about. For example, when Gene puts on Finny's ridiculous pink shirt and looks in the mirror it's the most obvious example that he wants to 'be' Finny. This is a clear metaphor of his basic motivation.
But the bottom line is that it's a good story that makes us squirm, giggle, and cry.
I would argue what makes this an incredibly important work about literature is how the novel is used to comment upon war and in particular the tendency we have to create inner enemies. Although the novel is set during wartime and war occupies the thoughts of so many characters, there are no shots and no deaths related to war. Instead, the novel focuses on wars fought internally, within our hearts. This kind of internal war is part of growing up, the novel suggests, as we all go through a stage when we live in a hostile world and that it contains some form of enemy that we need to fight against. War, in this sense, is related to the real war in that it is seen as being equated with the loss of childhood innocence. Each of the boys in this novel respond to this loss of innocence in a different way. This of course results in one of the most famous quotes of this novel:
I never killed anybody and I never developed an intense level of hatred for the enemy. Because my war ended before I ever put on a uniform; I was on active duty all my time at school; I killed my enemy there. Only Phineas never was afraid, only Phineas never hated anyone.
Gene thus realises that he created his own enemy that he fought against, and that his enemy was actually his own jealousy and envy of Finny. All humans set up their own enemies and fight against them, this novel suggests, except for Finny, who, because of his innocence, is unable to comprehend how anybody could be his enemy. Finny dies because he is to innocent for this world.