Continuity mistake: While ET is lying on the bathroom floor there's a pink and burgundy striped bath rug beside him, and in some shots the burgundy edge is beside him, but in other shots the rug is turned around and positioned the opposite way with the pink edge beside ET. Additionally, the proximity of the rug to ET keeps changing during the scene. (This rug can also be seen during an earlier scene with Elliott and ET in the bathroom).
Continuity mistake: When Elliott gets food for himself and ET he takes the milk container from the top shelf of the fridge, and we see an empty space beside the Skippy peanut butter where the milk had been, but after Elliott drops everything onto the floor, in his next two closeups note the milk container (it has a distinctive curved design) is back on the shelf in the same spot beside the Skippy jar.
Continuity mistake: After ET weighs himself on the scale there's nothing draped over the side of the bathtub, but in the next shot we see in the reflection that there's a burgundy tub mat hanging over the side of the bathtub, behind ET and Elliott. Then that tub mat vanishes again when Elliott rushes to answer the telephone.
Continuity mistake: When Elliott finds ET hiding behind the stuffed animals in his closet, Elliott's holding two plates in his hands while balancing two cans of Coke on his forearms, then one of the cans falls over onto the plate, but in the next shot we can see that can under the plate and it's upright.
Continuity mistake: When ET is trying to tell Elliott and Gertie that he wants to phone home there's a chip bag on the table in front of the turntable, but it disappears when Michael comes home and ET points out the window and says, "ET phone home," then the chip bag reappears when ET pulls off the wig.
Other mistake: While the kids are huddled around the kitchen table, Elliott is saying his lines, while Gertie is actually moving her lips, saying those same lines on the right side of the screen.
Continuity mistake: When Elliott's holding the Kromedge saw blade in his hands, note the position of the writing on the blade and the position of his fingers within the blade's teeth, then after he slices the index finger of his left hand there's a closeup of the blade with Elliott's blood on one of the teeth. The problem is that Elliott's index finger was actually on the blade tooth further along the blade's circumference (note the writing).
Continuity mistake: After ET levitates the balls of modeling clay (which Gertie had been playing with) Elliott tells Michael "something scary" is happening, and when Elliott runs toward the shed the swings on the swing set are motionless, but in the next shot one of the swings is suddenly swinging back and forth, but in the next shot it's perfectly idle once again.
Continuity mistake: After Elliott finds ET all dressed up courtesy of Gertie, when ET heads into Elliott's bedroom the large oval striped rug has its center under Elliott's bunk-bed, but when they're all in Elliott's room that rug is pulled out (which can't easily happen due to the weight of the bunk-bed) and now covers much more of the floor area, between the bunk-bed and closet.
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.
Chosen answer: They actually are smoking and there are cigarettes on the table.
Mike K