The primary and most obvious difference between pulled pork and brisket is that they come from two different animals - a pig and a steer respectively.
Aside from that there are a number of other differences like cost per serving and ease of smoking (and everything that encompasses).
At the end of the day, it more or less boils down to personal preference and if you like beef or pork more.
As the name suggests, pulled pork comes from a pig. The two most popular cuts that are used for pulled pork are the shoulder picnic and the pork butt.
Note: These two cuts have their own set of differences. I have an entire article that breaks them down, found here. The short of it is, Pork butt is better used for pulled pork simply because it contains more intramuscular fat.
Boston butt comes from the upper shoulder of the pig, behind the head.
The meat is smoked until roughly 195 - 205F internal, at which point the shoulder muscles can quite literally "pull apart" - hence pulled pork.
Pork butt that's tender enough can be pulled apart simply using your hands. Granted, the meat might be too hot to do so. In which case people use forks, meat claws, or they use nitrile gloves with cotton liners (to protect against the heat).
The brisket primal is a cut is from the breast section of a cow under the first five ribs.
Brisket is best described as the toughest cut of meat from the cow - while at the same it's considered the best cut of beef you can smoke.
Since cows don't have collarbones, much of the cow's weight is supported by the brisket. As a result, a lot of connective tissue - primarily collagen - develop.
The brisket primal is made up of two muscles - or sub-primals - the flat and the point. These muscles are separated by a thick layer of fat; The flat is oriented above the fat and the point is oriented below.
You can learn about the difference between these two muscles in this article.
In every case, pulled pork is cheaper than brisket.
In terms of cost, pork butt is 2.5 - 4x cheaper per pound.
Both are also similar in terms of yield; Brisket and Boston butt will typically have a 50-60% usable meat yield.
In my opinion, brisket is way more difficult to smoke than pulled pork, in just about every aspect of the smoking process.
Whether it's selecting the meat, trimming the meat, and then smoking the meat.
One of the biggest reasons pork butt is easier to smoke than brisket is because it's hard to pick to a good brisket and it's easy to pick a pork butt.
Beef is graded using the USDA grading system. For all intents and purposes, you'll want to work with at least a USDA Choice brisket.
The main thing that separates USDA Grades is the marbling or the fat that's dispersed throughout the lean meat.
With pork butt, there is no grading system simply because:
"...it is generally produced from young animals that have been bred and fed to produce more uniformly tender meat."USDA.gov
Essentially with pork butt, the only thing that actually matters is the weight of the meat. Beyond that, some folks like boneless pork butts while others prefer bone-in pork butts.
Some people like to leave the fat cap on, others completely remove it. Some People opt to remove the false fat cap and leave the actual fat cap on.
With brisket, aside from using USDA Choice grade or higher, I also look for things like:
Trimming brisket is a rather nuanced processed - so much so that I have a really long guide that goes over my process.
Brisket trimming is to the point where the barbecue world has created their own nomenclature for the different parts of the brisket itself.
I'd strongly suggest checking out my trimming guide here.
Essentially the goal with trimming brisket is to maximize edible meat and to make the brisket more aerodynamic in the smoker.
With pork butt, there is minimal effort - if any - required for trimming. Personally, I don't tend to do any trimming at all. Sometimes I'll remove the false fat cap but nothing else beyond that.
The false cap is on the "fat cap" side of the pork butt. It's essentially what appears to be the fat cap but is actually a thin layer of fat, followed by a thin layer of meat.
Below that is the actual fat cap.
This false cap can technically prevent the pentation of your dry rub and in some cases be inedible.
Some folks opt to completely remove the fat cap too.
The reason this is possible is because there is so much intramuscular fat that the fat cap isn't necessary for "shielding" the meat like with brisket.
However, on my Pit Barrel Cooker where I'm cooking butts at 325F+, having a false cap to shield the meat works well. Typically the false cap will peal right off and you're left with succulent fat to mix with the rest of the meat.
The sandwich could feature any number of things like a traditional coleslaw, pickled red onions, pickles, etc.
Aside from a sandwich, pulled pork can be used on top of nachos, tacos, mac and cheese, BBQ pizza, or even eaten on its own.
Brisket - both the point and the flat - are sliced against the grain and served as slices.
At a barbecue restaurant you're usually given several pieces of white bread to eat the brisket slices with. However, you can also simply eat brisket as it is in slice form.
Brisket point can also be made entirely into "burnt ends." Some folks opt to separate the flat and the point in the middle of the cook (pictured below) and cube up the point into small squares.
Burnt ends are the best part of the entire brisket. When smoked whole you only get two sides from the point muscle to make into burnt ends.
I've also seen brisket added to dishes like Mac and cheese, quesadillas, tacos, chili, beans, nachos, hash, chopped brisket sandwiches, etc.
This is super subjective and really boils down to your own personal preference.
For example, I quite like beef and would rather have a brisket over pulled pork. Where-as my father is the complete opposite, he'd much rather have pulled pork over brisket.
Considering they both take around the same time to smoke (8-12+ hours) and rest (2 hours) I basically decide if I feel like one vs the other.
Granted, I don't really have a great local source for brisket where-as I can get Pork Butt from BJ's Wholesale store year round.