Carrington, who tore a quadriceps tendon in last Sunday's 27-20 loss to the New York Jets and has since been placed on Injured Reserve, played a very specific role in Mike Pettine's defense - he was a power end, capable of playing four- and five-technique or sliding inside on passing downs. That type of player needs range to set the edge, but enough power to hold up inside. Carrington will not be easily replaced.
Branch measures in at 6'6" and 325 pounds. He played some four-technique while with the Seattle Seahawks, but is not an athletic fit for that power end role - he lacks the range to make plays on the perimeter, and is more of an interior lane-plugger. That may be why the Bills are talking up the 6'4", 300-pound Bryant - he has a bit more pep in his step, and offers a closer approximation to Carrington than Branch does.
Bryant was inactive in Week 1, and has played 37 snaps between the Bills' last two games. Branch has 110 snaps to his name on the season, and is clearly the more experienced of the two players. You'll likely see both players on the field more often with Carrington out of the lineup, but in base sets, expect Bryant to work with the first team.