Trim most of the visible fat from the outside of the pork butt leaving about 1/8- to 1/4-inch layer of fat on top. Place pork on baking pan. Rub all over with Sweet and Smoky Rub. Cover with plastic wrap. Let stand at room temperature 1 hour.
Prepare grill for indirect medium-low heat (250°F to 325°F). Preheat grill by turning all burners to high. Turn burner on 1 side to medium-high. Turn off burner(s) on other side. Carefully place foil pan under grates on unlit side of grill to catch drippings. Place pork, fat-side up, on unlit side of grill. Close grill.
Note: To maintain medium-low heat (250°F to 325°F), keep lid closed and adjust lit burner as necessary. Directions were developed using a gas grill. Grills vary; cooking time is approximate.
Grill about 6 hours until internal temperature of thickest part of pork butt is 165°F. Carefully remove pork from grill and wrap in foil.
Return pork to grill, placing it back on unlit side. Grill 1 to 2 hours longer or until internal temperature is 195°F to 200°F or until meat is tender and bone twists easily.
Remove pork from grill. Let stand 15 to 20 minutes. Slice or shred with 2 forks. Serve with warm pan juices or your favorite barbecue sauce, if desired.
Smoking Tip: Soak 1 cup hickory wood chips in enough water to cover for 1 hour. Drain. Fill smoker box with wet wood chips. (If your grill does not have a smoker box, place wood chips on a large sheet of heavy duty foil. Fold up sides to form a pouch. Poke several holes in foil.) Place smoker box under grill grates on one side of grill. Close lid. Continue as directed on Step 2. The side of the grill with the smoker box should be the lit side. Refill the smoker box with fresh wet wood chips every 1 1/2 to 2 hours.