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)
Other mistake: When Elliot and ET finally meet in the backyard at night, ET returns some of the candy that Elliot had left for him. In the shot before ET places the candy before Elliot, we see him standing with all his fingers extended, so he is definitely not holding any candy in his hands. He is also not wearing clothes, so he has no pockets. Where was ET keeping the candy before giving it back to Elliot?
Continuity mistake: In the scene on Trick-or-Treat night, Elliot and E.T. go to the forest so that E.T. can set up his communication device to "phone home". While they are there, Elliot is appealing to E.T. for him to stay, saying he will get used to it. While he is talking to E.T. he is inches away from his face. But in the very next shot it shows E.T. looking around, staring into space, with Elliot nowhere to be seen. But the next shot shows Elliot is in E.T.'s face again just as he was two shots before.
Continuity mistake: Throughout the movie E.T.'s fingers are shown to be very long, thin, and boney. But in the final scene, when Elliot and E.T. are saying goodbye, E.T. reaches down to pick up the flower pot to take with him. The camera shows his hands picking up the flower pot but they appear the size of human hands wearing gloves, not the long skinny fingered hands we saw during the entire movie.
Factual error: The government operatives go to great lengths to cordon-off and sanitize Elliott's home with extensive hazardous material preparation. Controlled airflow, Hazmat suits and everything. But as soon as E.T. "dies," everybody takes off their Hazmat masks. So, E.T. was only a Hazmat threat when he was alive? A real government Hazmat team would never drop their guard like this, as long as the alien body was still present.
Visible crew/equipment: When Elliot is luring ET with Reeces Pieces and goes into his room and through his drawers to find more candy, you can see the shadow of the crew moving on the wall next to Elliot.