I would say it's both Carlo's wanting to get back at santino AND Barzini blackmailing him.
I bet Barzini blackmailed him because he was doing something naughty and thretened to expose him..probably some dirty business in his bet placing job he got. I dunno.
But I was sad to see carlo killed in the book.
I was happy to see him go in the movie. Because the book kinda shows y'know Carlo's side and stuff.