Jimmy was a childhood friend of Butch Gilzean, and the two ran together even "back in the day". Jimmy joined the Falcone Crime Family and moved up the ranks, eventually becoming a underboss. Jimmy affectionately referred to Butch as "brother". While Butch feigned reciprocation, he was secretly willing to take down his old pal when it suited him.
He technically had seniority over Fish Mooney, so when her scheme to use Nikolai to off the old man fell through, she turned to Jimmy for support in a coup. Since Jimmy had ambitions to sit in the big chair himself, he was less than receptive, dismissing her proposals and retorting that he had seniority, insinuating that she better back off. In fact, he turned to his old buddy Butch and tried to enlist him in removing Fish from the picture. Having weighed his options, Butch decided he was better off with Fish, and killed Jimmy himself, after an apology for an incident in teenage when he stiffed his friend of his share of a robbery.
Jimmy was vocal, confident, and not afraid to voice his opinions. This was especially visible when he felt comfortable voicing his concerns in front of Falcone about raising their tariffs, and outwardly stating the need to oust him in front of Butch and Fish. An experienced mobster, he also felt confident in his skills of persuasion, believing that Butch would easily value him as an ally over Fish, because of his connections and status. He was great at being a "friend" to others, especially his own men, allowing Butch to "name his terms" as a member of his new order of rule. On paper, Saviano had more to offer than Fish, as he was more respected in the Falcone family. Unbeknownst to him, this was why Fish was motivated to get rid of him, while he still believed that Butch would easily choose their brotherly friendship over his fledgling relationship with Fish. Fish was able to convince Butch that she was more valuable of an asset than Jimmy, so Jimmy was subsequently disposed of by Butch.