For the rub: In a small bowl, stir together the brown sugar, paprika, salt, garlic powder, cumin, oregano and black pepper. Rub into the pork butt and allow to sit at room temperature, 1 to 2 hours.
For the wet mop: Meanwhile, in a medium bowl, whisk the apple juice, vinegar, mustard and pepper flakes together with 1 cup water. Cover and store in the refrigerator until ready to use.
Prepare a grill for indirect low heat (if desired, add wood chips to the burner element on the direct side): For gas grills (with 3 or more burners), turn all the burners to medium-high heat; after about 15 minutes, turn off one of the middle burners and turn the remaining burners down to medium. For charcoal grills, bank one chimney starter-full of lit and ashed-over charcoal briquettes to one side of the grill. Set up a drip pan on the other side to avoid flare-ups. (Be sure to consult the grill manufacturer's guide for best results.)
Lightly oil the grill grates. Cook the pork butt over indirect heat, keeping the temperature consistently at about 225 degrees F, 5 hours and 30 minutes. Continue to cook, basting the pork butt with the wet mop sauce every 20 minutes, until the meat reaches an internal temperature of 195 degrees F, 2 hours and 30 minutes to 4 hours and 30 minutes more. Transfer the pork butt to an aluminum pan, tent loosely with foil and allow the meat to rest 20 minutes. Use 2 forks to shred the meat and transfer to a bowl. Add drippings and any leftover mop sauce to the pulled meat to moisten further.