Visible crew/equipment: When Michael and Elliott bring ghost ET downstairs Mary walks over to the trio with the Polaroid camera, and when she kneels down the actor's tape mark can be seen on the carpet, presumably where Mary had stood when she stopped to scream.
Other mistake: While ET is pulling the blanket loaded down with small appliances, etc., the TV is playing "Francis Goes to the Races" and when the film's Peter and Frances walk into the stall we hear Frances (the woman, not the mule Francis) say, "Grandpa, what's wrong?" but the subtitles on the DVD (1982 and 2002 versions) read, "Stanford, what's wrong?" and the subtitles on Netflix (1982 version) read, "Francis, what's wrong?" Additionally, when ET pulls the blanket behind the TV, on the TV screen we see Frances join her grandfather and we hear their dialogue, but in the next shot as ET stumbles out from behind the TV very drunk, even though we hear the TV's continued dialogue, what we actually see on the TV screen is a repeat of Frances joining her grandfather again, and then it jumps to the scene of Peter in the stall with Francis.
Continuity mistake: When Michael drives the van onto the playground and stops, in the shot from behind the three boys on their bikes Greg turns his head to look down at Harvey, who's sitting between Greg and Steve, but in the next shot facing the boys Harvey has vanished.
Continuity mistake: When Mary sees ET lying on the floor in the bathroom, in the closeups of Gertie there's a white fringed hand towel hanging on the rod behind her, but in the wideshots that towel isn't on the rod though we see a small white towel lying on the counter when Michael picks her up. Then in the following shots of ET the white fringed hand towel is hanging at the end of the rod, but then it vanishes.
Continuity mistake: When Elliott finds ET dressed up and wearing a wig both sides of Elliott's closet doors are open, but after ET hands Elliott the newspaper comics section ET goes over to the closet and only one side is open. Additionally, note the deliberate absence of Elliott's lounge chair which should be right beside the closet door.
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 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.
Visible crew/equipment: After Mary and all the boys look in the shed and find tracks, when they head back to the house Tyler stops at the pizza box, and note the three distinct marks on the ground to inconspicuously signify where actors should stand while discussing the pizza. Michael stands at his curved mark, Elliott stands at his L-shape mark, and Mary's mark is in the shape of a T-mark.
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: 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.
Revealing mistake: Elliot takes his dishes to the sink, opens the faucet and steam rises all over the kitchen. When the camera pulls back, one can notice that the huge cloud of steam is coming from somewhere along the window, not from the narrow stream of water from the faucet.
Chosen answer: They actually are smoking and there are cigarettes on the table.
Mike K