It goes back to Frank Costello and the Gallo brothers. They avoided each other much of the time, but by the late 1970s there was much dislike for each other. Persico was sending guy to other families outside NYC to be made, then transferred back to the Colombo family, Carmine also was backing up guys in his family that was swaying influence in other families that the Genovese family had strong ties and interests with. There is also Philadelphia and Bonanno families situations, before the Chin could vote, where Persico went against most of the other families. The reason why the Chin was supporting Persico in the 3rd Colombo war were 1) The Genovese believed John Gotti was manipulating Orena to gain more power and money for himself and the Gambinos. 2) Gigante was still gunning for John Gotti. 3) LE was cracking down on everyone to where logically it would be better to stabilize that family.
Bottomline, it was who were the Genovese family better off with in business dealings and voting on the commission.