John Stanfa wanted John Veasey dead for a couple of reasons, really. After Veasey murdered Frank Baldino (an associate of Joey Merlino), Veasey allegedly began to attempt to extort money from other mobsters, because by this time Veasey had an inflated ego. Secondly, Veasey claimed in the 60 minute interview that Stanfa was going to give him a cash bonus for being inducted into La Cosa Nostra. Veasey went to confront Stanfa about it, Stanfa refused, and wanted John eliminated (it goes with the idea that you don't take money from the family, let alone the boss!). In other words, John got too big for his britches.

As for Billy Vaesey, the mob subculture is a small world, so maybe he got word that Stanfa was plotting to kill his brother (Billy was friends with Joey Merlino as a kid, so word probably gets around really fast).