Birria originated in Mexico and barbacoa is thought to have originated in the Caribbean, but over time barbacoa has become a popular cooking style in Mexico.
Birria has a thinner consistency than barbacoa, so it's a little soupy and traditionally served as a stew. These days, you'll find it in tacos and even on nachos.
Barbacoa lacks the broth that birria has and more closely resembles U.S. barbecue. Both cooking methods involve slow-cooking the meat until it's fall-apart tender.
Birria is typically cooked over a flame, though a slow cooker also works. The beef is braised in a consomé (made with water or beef stock) for several hours until tender.
The entire broth mixture is blended until smooth, then both the beef and the consomé are added back into the pot. It can be served as a stew or on top of tacos and other dishes.
Barbacoa is cooked in a large oven dug into the ground and usually made from bricks; the oven is preheated using wood until it's piping hot. The meat is wrapped in maguey leaves.
This crucial part of the cooking method adds flavor and locks in the meat's moisture. The pot is lowered into the in-ground oven and covered with the organs from the same animal.