He might've been younger and more astute, but he overplayed his hand by making clear it was he who was behind the War all along by being so insistent about the drug trade. Basically gave himself away in much the same fashion as Sonny gave away his interest in the drug trade which led to Vito being shot.
But by the time of the Dons' meeting, Barzini had already accomplished his goal so much so that he didn't need to hide himself anymore behind Tattaglia. Also, he needed to give a signal that he was the new power by chairing that meeting.
