Continuity mistake: When Gertie rushes into Elliott's bedroom with the drawing she made for him, just as she sees ET and begins to scream the paper is in her left hand and her right hand is empty, but in the next shot the paper is suddenly in her right hand.
Continuity mistake: When everyone goes to the shed Michael kneels down to look closely at the prints on the ground, and in the closeup the marks we see differ from the clear prints seen in the next wideshot.
Continuity mistake: After Elliott finds ET in the wig and hears him speak, when ET gives Elliott the newspaper comic section there's nothing on the yellow chair behind him, but in the next closeup a shirt suddenly hangs over the back. Additionally, the globe on the desk has turned so the continents change position.
Continuity mistake: When ET tastes the potato salad in the side shot the drawer in pulled open, but in the shot from behind him as his finger touches the can of beer we see the drawer under that shelf is closed, but in the next side shot that drawer is open again.
Continuity mistake: After Elliott explains to ET how the water comes out of the bathtub faucet, when Elliott's on his knees moving closer to ET (while he's squeezing toothpaste), note there are two solid color floor mats (one white and one brownish) which disappear when Elliott rushes to answer the telephone.
Continuity mistake: In the first shot of the boys playing Dungeons and Dragons, Greg is holding the phone with the stretched out cord pulled under his left armpit, then a couple of shots later (not a continuity issue yet) the cord is pulled over Greg's right shoulder, but in the next shot it's back under his left armpit.
Continuity mistake: While Elliot and E.T. are saying goodbye to each other at the spaceship, you see Elliot's mom sink to her knees. Then, you see a close-up shot of Elliot and E.T. looking at each other and you can see her white shirt in the background as she gets back up, but then there is a close up of her face as she gets up again.
Continuity mistake: While all the boys are trying to escape with ET, there are black brake levers/clamps on Michael's and Greg's bike handlebars, and brass brake levers/clamps on Tyler's and Steve's handlebars, but in some closeups this changes. For example, after Steve shouts, "Let's split up," Michael's are brass, when Tyler pulls off his balaclava his are black, and when Greg shouts, "Tell me when it's over," his are brass. Presumably bikes were switched for closeups.
Continuity mistake: When Elliot's mother is retrieving a heavy blanket to put over him because he is faking an illness, she tears open the closet doors and the leading edges of both doors come free from their tracks. A moment later when the shot angle changes, and you see her from inside of the closet, both doors are open, but both are back on their tracks.
Continuity mistake: The street the boys turn onto during the chase, when the boy says "we made it," isn't the same street as they had turned into. There should have been houses, but instead there were just trees and cars.
Continuity mistake: The size of the eye holes on E.T.'s ghost outfit change shape and size several times (most noticeable when Michael finds the abandoned outfit when he is looking for E.T.).
Continuity mistake: When Elliot is leading E.T. up to his bedroom for the first time first his shadow is in front of him on the candy, then in a reverse shot it is behind him.
Continuity mistake: When Elliott slices his finger on the Kromedge saw blade, it's the index finger of his left hand, but in the closeups of his finger he's actually holding up the index finger of his right hand.
Continuity mistake: After Mary carries Elliott downstairs the government agents start entering her home, and in the next closeup of ET lying on the floor there's a large bath towel hanging over the side of the bathtub, but in the next shot of ET from behind the "spacesuit", there's now a small bath mat (same color as the previous towel) hanging over the side of the tub.
Continuity mistake: When Mary and the boys go to the tool shed after Elliott's encounter, as Tyler and Michael kneel down to inspect the tracks on the ground, the trash can beside Tyler is partially under the table and a roll of fencing is upright behind him. However, after Michael mentions the coyote, the trash can is now to the left of the table, the roll of fencing is lying on the ground in front of the trash can, and some long-handled tools have moved as well.
Continuity mistake: When Elliott is trying to lure ET into his room by leaving Reese's Pieces on the carpet, the red helmet is lying on the table with its white stripe facing the doorway and its opening facing the bunk-bed, but when ET reaches the doorway the helmet has turned itself around so it's facing the opposite way.
Continuity mistake: When the two government agents pull over at the bottom of the hill to cut the boys off, in the shot from inside the car we see Greg and Tyler riding their bikes over the car and zooming off, then the agent inside the car lifts his head as yet another bike lands on the car, but in the next exterior shot it's Tyler (note his black high-tops) who lands on the car's roof, knocking off the magnetic beacon light. (Elliott's conspicuously missing from these few shots).
Continuity mistake: Many of the night scenes take place in the back yard, or in the cornfield which is shown to be behind the back yard. Daytime shots show the house to be in the suburbs, on a hill, with no cornfield or possible space for a cornfield.
Continuity mistake: When E.T. Is at home with the speak and spell, the back cover is missing as he is working on it. In the next shot, the back cover is on it again.
Continuity mistake: While ET is watching Mary read the story of Peter Pan to Gertie, in the shot from outside the closet we see Elliott using his left hand to close the shutter slats, but in the next shot inside the closet it is Elliott's right hand closing the slats.
Chosen answer: They actually are smoking and there are cigarettes on the table.
Mike K