Continuity mistake: When E.T. is in the kitchen reading comics, he glances over at the TV then takes a look at the coffee table which has, among other items scattered on top, six pack rings. A few shots later E.T. is at the coffee table working to create a communication device, but now there's one can of beer attached to the rings. (00:48:05)
Continuity mistake: When ET is walking with Elliot and his brother trick-or-treating, we see several shots from ET's point of view of people walking toward him. In all of the following shots however, none of the people that were walking toward him can now be seen behind him as they should be. It's not a montage either, these scenes are to be taken that they are occurring in real time.
Revealing mistake: During the scene in the woods with the communication machine, Elliot tells E.T. that they need to go. You see the machine still working at a regular interval because a rope is pulling it. The shot pans to the left a touch and the rope stops moving, as if the person pulling it realises it's no longer in the shot.