Gene and Finny are indeed best friends, though their personalities make them an odd couple. Gene is reserved, studious, and lacking in self-confidence while Finny is charismatic, athletic, and popular. More significantly, they view the world in fundamentally different ways. Finny is trusting and does not question the motives of others, nor does he seek their approval. When he breaks the school swimming record, for example, he asks that Gene not tell anyone. Gene, on the other hand, is jealous of Finny's popularity and especially his ability to flout rules and his seemingly effortless success at athletics. He assumes that Finny must be jealous of him too, and that his attempts to pry him away from his studies are really motivated by a desire to ruin his grades. It is when he realizes that Finny is not, in fact, jealous of him that he jiggles the tree branch, forcing Finny to fall to the ground. Finny, it seems, is more of a true friend than is Gene, though Gene's lifelong feelings of guilt suggest that he cared deeply for his friend.
I believe that at one point in time they were close but as time drags on Gene slowly loses his mind and creates reasons to hate Finny. Gene is actually jealous of Finny for being such a people person, and a great athlete.