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.
Factual error: Unbelievable that for 30 years no one has pointed out that in every shot with the moon, except when the moon is full, the crescent is pointing the wrong direction. In the evening (ordering pizza), in the Northern Hemisphere (California), a crescent moon will always be pointing down to your right (the bright side will be on the lower right). The only time it could point the way it does in these scenes is in the wee hours of the morning, at 4 or 5 a.m. and looking to the East. Spielberg didn't even fix it in the re-release.
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.