Gene does not cry at Phineas's funeral because he feels it is his own, and you don't, he says, cry at your own funeral. He experiences it as his own funeral because Finny felt like a part of himself: they were very close, and Gene believes he will never meet with his friend's generosity of spirit again. Phineas made Gene feel seen and understood as nobody else did.
For example, when Gene visits Phineas right before his surgery, he confesses that he purposely bounced the tree branch so his friend would fall. When he tries to explain it was the impulse of a moment, Finny understands what he is trying to say. Finny says:
I believe you. It’s okay because I understand and I believe you. You’ve already shown me and I believe you.
For Gene, Finny was a special person. As Gene says at the end of the novel: "Only Phineas never was afraid, only Phineas never hated anyone."
Many of us can understand how Gene feels—the feeling he expresses is universal—because almost everyone has known a person who touches us more deeply than other people do. For Gene, that person was Finny. If we are, in part, how other people see us, the part of Gene that only Finny saw is now gone forever.