A whole brisket is just that - the entire brisket muscle including both of the muscles it's made of, colloquially called the point and the flat.
A flat brisket features only the pectoralis profundi muscle. In this case the point or pectoralis superficialis muscle is trimmed/removed from the brisket.
A whole brisket refers to the entire brisket primal; The brisket primal is made up two muscles - when these muscles are separated, they are called "sub-primals."
The muscles that make up the brisket are the pectoralis profundi and the pectoralis superficialis. Colloquially, these muscles are called the flat and the point, respectively.
These muscles are separated by large seam of fat. With the fat cap facing down, the flat muscle is oriented above this fat seam and the point is oriented below it.
Cutting along this fat seam entirely separates the muscles.
A whole brisket is also commonly called a "packer's brisket" or a "full packer."
The term "Packer" simply refers to the brisket muscle being packaged whole - bone removed and untrimmed - in a meat processing/packing plant.
When looking at the USDA Institutional Meat Purchasing Standard (IMPS) - the brisket you'll find at your grocery store is "120 Beef Brisket, Deckle-off, Boneless.
The IMPS defines it as:
"This item is as described in Item No. 119 except that the deckle (hard fat and M. intercostales interni on the inside surface) must be removed at the natural seam exposing the lean surface of the M. pectoralis profundi. The hard fat along the sternum edge must be trimmed level with the boned surface. The inside lean surface must be trimmed practically free of fat."IMPS
Note: From the IMPS documentation we also know 120A is Brisket Flat and 120B is Brisket point.
Funny enough, not many people even realize that brisket does have bones attached to it. In my 29 years of life, I've never seen a brisket sold in a grocery store as bone-in.
Even asking my father - he said never in a grocery store but from local butchers (which also don't exist in our area anymore) you could order them.
The main reason bones are removed is due to the advent of cryovac/vacuum sealing technology in the 70s. The bones of the meat would tear the plastic and render the vacuum null.
The bones from the beef brisket are also rather easy to remove and since barbecue is quite popular now, it's sold whole, boneless, and deckle-off.
Brisket flat refers to the pectoralis profundi, or the deep pectoral of the brisket primal.
Other names People use to refer to brisket flat:
The term "flat" is a great descriptor for the muscle. It's quite literally a flat, rectangular piece of meat.
In a grocery store, it's not uncommon to find brisket flat separated from the point. The flat makes up the majority of the beef brisket and weighs anywhere from 3 - 10 lbs.
It really depends on your use-case and what you like.
If you're smoking the meat, I find brisket flat tends to dry out and isn't great for smoking. Rather, a whole beef brisket works great for smoking - especially if you buy Prime or Choice grade brisket.
However, if you're looking to make brisket in a slow cooker like a crock pot, it's unlikely that you'd fit an entire brisket. Where-as a brisket flat works perfect, especially when cooked in a liquid for an extended period of time.
Brisket flat sold without the point is quite popular. A common example is corned beef that's sold in a pre-packed brine solution around St. Patrick's day.
Most folks who do barbecue actually prefer the point (the fat end), rather than the flat (the lean end). To people who don't know barbecue, I'd wager to say they are more aware of the flat than they are the point.
Truth be told, they may not even know it's made up of two muscles - which are also sliced in different directions.
In my small town in New Hampshire, the flat is what most people expect when they hear beef "brisket." My local grocery store doesn't sell or cut packer briskets because they have to meat Consumer expectations (quoting one of the meat cutters).