Excercise ROM:
Your lats pull on your upper arm, the humerus. If you have a shoulder-width grip, during a pull-up, your upper arm travels from pointing roughly straight up, to pointing roughly straight down, a full 180 deg.
Whereas if you have a wide grip, your upper arm is not pointing straight up in the hang phase, its probably somewhere more around 30 deg short of vertical, so your upper arm is traveling only about 150 deg to point straight down at the top of the lift.
So wide grip does not have the same range of motion for your lats as shoulder width. However, it does decrease the leverage of the biceps and the lats, so you are making the lats work harder. Another thing it does is force you to keep your elbows out to the side more during the pull-up, instead of coming in front of you. Those things combine to make a BW pull-up harder on the lats. Of course you could instead do a regular shoulder-width pull-up and just add mroe weight instead.
Essentially, using a wide grip pull-up to target your lats is sort of like using a quarter squat to target your quads: you’re able to load up more on the weight the muscle is lifting and the muscle is working a lot harder, but through a decreased range of motion.
Rotator cuff health and ROM:
Think of your rotator cuff like a cup around the top of the humerus, facing forward and slightly down. If you raise the arm to far above the head or out to the side, your upper arm is reaching the edges of the cup. The way your body allows more range of motion is by pulling the shoulder blade down and in, sort of opening the rotator cuff up and out, to allow for your arm to travel further up and back.
So wide grip and BTN pull-ups just require you to pull your shoulder blades in and down to open up the rotator cuff “cup” and allow your arm to go out wider. I think they both give you a bit of work in the upper middle of your back because your Rhomboids probably are working to pull your shoulder blades in and open up your rotator cuff for more upper arm ROM.
While this extreme stretching of the rotator cuff is probably fine for people with a healthy cuff, it might cause problems if you’ve got any rotator cuff issues.