Henry is often overlooked, he was regarded as Roy's best shooter.
Though he made a mistake when he asked Roy for permission to kill Chris.
Henry was pissed that after the Andrei Katz murder when he and Joey went to jail, Chris never provided his (Henrys) family with any money.
I assume Anthony cared for Joey's family?
Henry wanted to clip Chris but Roy went berserk as you might imagine.
Also, After Chris's eventual death, Joey became Roy's number 2 man. Henry didn't get the job because he used to hang with Dom in bars and Roy didn't want Henry telling Dom things incase he passed them on to Nino.