Continuity mistake: When Mary gets her first look at an ailing E.T. lying on the bathroom floor, she panics then hurries everyone downstairs - leaving E.T. behind. Moments later one of the astronauts happens upon E.T. and now a bright pink bath towel can be seen draped over the bathtub's edge, which wasn't even there to begin with. (01:16:00 - 01:17:55)
Continuity mistake: When Elliot burps, the pencil on his desk disappears between shots. Also, the paper swaps from laying on the edge to several centimeters away.
Other mistake: When the kids pedal away from the FBI and turn towards the homes under construction, skidmarks from previous takes are visible before the FBI cars arrive.
Continuity mistake: When the kids and the mother go outside to see what scared Elliot, ET's footprints on the sand are smudged. When the shot changes they're neatly marked.
Continuity mistake: At the end, ET's finger glows and he says "I'll be right here" but in the immediate opposite angle the finger's not glowing.
Revealing mistake: When ET is grabbing things from the fridge, in the close-up his fingers swap from thin and long to chubby and short, obviously a puppeteer's glove.
Continuity mistake: When ET's spaceship takes off and starts flying, the sky is red, a shot later it's blue.
Chosen answer: They actually are smoking and there are cigarettes on the table.
Mike K