What Obsidian said, but also, I think to some extent Sephiroth never got the emotional encouragement Angeal or Genesis did while being in SOLDIER.
Angeal is very down-to-earth, thinking of his father and the importance of protecting those around him. His finding out he wasn't human was just as shocking as Genesis' reaction, but he somehow maintained composure through Zack's steadfast friendship alone.
Genesis is the first to realise he's not human and at the same time realise he is also dying...growing up on a farm the way he did, reading books and such, I can only assume his pleasant life was shattered. He felt he needed his friend Angeal to shoulder his turmoil...and probably spoke to him many times when the two left SOLDIER leaving Zack behind.
What's more, when Zack finally has some semblance of what's going on, he actually begins to converse with Genesis and understands his desperation (reading between the lines of the poem to his desperate cries for help). Not only is Genesis dealing with his physical condition deteriorating, he's at odds with Shinra Company who are hunting him down.
As if that weren't enough, Genesis has a very cocky "hero streak" in him that relates to his dream to actually become a hero (this is very similar to Cloud's dream actually)...when he fights Sephiroth, it's very heated and hot-blooded, to the point he considers Sephiroth his absolute rival (even compared to his friend Angeal). To me it's strange that Angeal doesn't view Sephiroth this way at all, yet Zack still idolizes Sephiroth in some small way (even Cloud did, based on what he knew of him).
That sense of rivalry escalates even more when the truth slowly reveals itself. Genesis isn't just trying to find the truth, he's trying to find meaning in his life...even if that meaning is muddled in a Loveless poem. He still WANTS to be a hero (this relates to Zack's "embracing your dreams"), despite his circumstances...so when he discovers JENOVA, part of what he says to Sephiroth is Spite (due to his rivalry), part of it is that he simply wants a future, (which for some reason Sephiroth flatly denies him...but actually comes out much later when Sephiroth claims he is to "inherit" the power of the Ancients and rule the planet, in other words, Sephiroth deeply distrusted everyone around him from the very beginning, unlike Genesis), and also, part of it has to do with seeking solace in the fact he isn't the only being with inhuman aspects...that is to say, he was HOPING Sephiroth would take his side and understand him like Angeal had...he was looking for friendship in his rivalry, but the truth, along with the spite, were part of the trigger that set Sephiroth over the edge.
Somehow further along the way he found comfort and truth from Zack, despite the chaos around him...which actually lead to Genesis amending his ways and actually finding a NEW path.
Contrast this with Sephiroth, and you realise just how lonely Sephiroth truly was...seeking solace in hatred for others, seeking spite which he unleashed on Cloud and even Tifa and especially Aerith. Even when Cloud put him down for the first time, he wouldn't go without a fight...and I think more to the point, Sephiroth was scared of being alone.
In Dissidia, Sephiroth keeps trying to give Cloud a reason to fight, when Cloud isn't the one who needs it...it's Sephiroth projecting his wishes on to Cloud. When Cloud flatly draws a line, it's Sephiroth who says with a rather fearful tone "Y-you don't exist--anywhere!" He was frightened of not existing, of not being real (he habitually taunts Cloud for being a puppet, but the real puppet was Sephiroth himself, a fabricated being!).
You could say right here, that the difference between Genesis and Sephiroth, is that Sephiroth didn't know he could find a place to be and was always wandering in the dark, relying on his instinct he inherited from JENOVA, and almost always the truths he discovered he distrusted and crushed, unable to accept his remaining human emotions and sadness or capable of trusting anyone...this culminates when he taunts Cloud once more, trying to crush what he cherishes...yet Cloud responds by saying "There is nothing [he doesn't] cherish".