The morning after the film's pivotal train wreck, Joe and Charlie are watching coverage of the story on local TV. The typeface on the screen is the now-ubiquitous Microsoft computer font Arial, most recognizable by its distinctive 'R'. However Arial was not invented until 1982 and not in broad use until after 1990. The movie is set in 1979.
Just after the train wreck, one of the boys describes the small block shapes they found in the trains cargo-hold as looking "like white Rubik's Cubes or something" (at around 1 min). Originally called the "Magic Cube" in 1977, the toy company Ideal® would not rename the toy "Rubik's Cube" until it was sold in the United States after May of 1980.
When Joe is in his room painting one of his monster models, you can see in the background a model of The Invisible Man(at around 21 mins). This model was not released until 2009 by Moebius models. Although the model was created in the classic Aurora models style, this model was not available during 1979.
When they show the map of where the dogs went (at around 31 mins), it shows I-675 as complete. I-675 was started in the early 1970s, but they only built it from I-70 to Fairborn. They resumed building I-675 a decade later, and was completed in 1987. The map shows it clearly going all the way from I-70 to I-75. It wasn't even close to reaching I-75 in 1979.
The sheriff (at around 45 mins) and his deputies (at 03:45 and 46:46 to 45:56) are shown driving white cruisers with blue/gray fenders. Up until the late 1990s, all sheriff's offices in Ohio used all black cruisers with white only on the roof and pillars with a 5-point star on the door.
In Joe's room is a model of the Space Shuttle and the external fuel tank is red/orange (at 09:35 along the right side of the frame in front of the Space Shuttle poster and again at 01:01:55). This change did not occur until the 80's as a cost saving measure. Shuttle models before this had a white tank.
The sheriff's deputies were all wearing semi-automatic pistols. In 1979 (the year the movie takes place) revolvers were by far the most common side arm of American law enforcement. While there are rare examples of law enforcement carrying semi-auto pistols during the 70s (the Illionois State Police adopted the Smith and Wesson model 39 in 1968), it was the mid to late 80s before semi-auto pistols became widely used, especially among smaller, rural agencies. Some agencies did allow for officers to carry personal sidearms, including the semi-auto pistols of the day, and you might see an occasional officer doing so, but this would not account for every deputy of a small, rural agency carrying them.
Joe is shown building an Aurora Hunchback of Notre Dame glow-in-the-dark model which is correct for 1979, but the model box is the "long box" used for the non-glow in the dark model in the sixties (09:24 to 09:30).
The patch seen on the Ohio Deputy Sheriffs did not come into use until 1996 or 1997 the patch before that was a triangular patch that said Deputy Sheriff with a 5 pointed star under it and the seal of Ohio inside that star.
After Joe and Charles board an evacuation bus (at around 1h 08 mins), a string of M35 2-and-a-half ton trucks pass one way and 5 ton trucks pulling red trailers pass the other way. The first M35 truck has the new central tire inflation system (CTIS) which was not put on M35 trucks until 1994 which is 15 years after the film's setting date of 1979. The CTIS can be readily identified by looking at the wheels on the truck. Notice instead of a typical basic lug system there is a large plate looking device over about a third of the wheel.
When the deputies are eavesdropping on USAF communications via radio, an airman can be heard saying Lima (as in the NATO phonetic alphabet). However, he pronounces it the wrong way, saying "LIE-MUH" (as in the Ohioan city), rather than "LEE-MUH", which is the correct pronunciation.
After the train wreck, one scene shows the car they came in has a steel or wooden beam/bar stuck into the rear wheel area and debris is piled close to the car - yet when they get in it to escape, the car is clear and they easily pull out of the lot (at around 35 mins).
When the boys break into a classroom at the school (at around 1h 14 mins), they break a window from a door and clear out all the glass. Later when the soldiers come in, the same door now has some of the glass still there (at around 1h 17 mins).
After the train wreck, Alice's car is dirty, has something sticking through the wheel/tire, and the trunk is up. But after they've all gotten into the car and she drives away, the car is perfectly clean, there is nothing sticking out of the tire, and the trunk is closed.
When Joe asks Cary to give him sparklers, he grabs some unlit ones (at around 1h 30 mins) and runs down a cave tunnel. In the next shot, the sparkler is lit. (This is an error only if one assumes Joe had no lighter or matches.)
In the closeup of the metal cube vibrating on the table in Joe's bedroom (26:48-26:54), the cube has a different design and finish than it does in any other shot. When Joe originally picks up that cube (at around 9 mins) and when he holds it while taking a bath (at around 29 mins), its sides are nearly flat. The vibrating design cube is also shown at 57:32 as Alice climbs into Joe's room and again when it vibrates a second time from 1:01:29 to 1:01:54.
When Joe picks up one of the cubes in a close-up (at around 8 mins), the cuff of his jacket conspicuously covers his hand up to the second joints of his fingers. When the scene cuts to a longer shot (at around 12 mins) the cuff is well back up his arm exposing his wrist.
When the kids get to the train station Martin's pages blow away and he chases after them (at around 45 mins). He is immediately seen in the next shot sitting on a bench studying his lines (at at 11:46). There was not enough time to get to the bench, whether he managed to get the lost pages or not.
At about 15 minutes-in, the kids arrive at the abandoned train station to shoot the 'parting scene' it appears to be very windy or gusty; yet when they start to shoot the scene the wind has all but gone.
The movie is set in southwest Ohio. A woman mentions in a town meeting (at around 49 mins) that "Belmont County's without power". Belmont County is in eastern Ohio. Vicky tells Deputy Lamb on the walkie-talkie that Brook County is out of power (at 43:29 using the spelling of Brook County from the captions). This is a fictitious county in Ohio, however principal shooting for the film was in Weirton, West Virginia, which is on the county line between Brooke County and Hancock County in northernmost West Virginia.
The 'four months later' scene (at around 7 mins) when the kids film their movie at the depot takes place after school is out - presumably late May or early June 1979. However, the TV in the background has Walter Cronkite talking about the Three Mile Island accident (at around 28 mins), which took place March 28, 1979.
The Lillian Airfield sign (at around 46 mins) shows "Town of Lillian, Ohio". In a later shot, the film shows another sign showing City of Lillian, Ohio. Ohio does not recognize any incorporated area as a town, only village or city.
The kids are seen riding 'BMX style" bikes. Many believe that this is a error, which is mainly because of misinformation. Since the early years of the 70's, children drew inspiration from motor-cross and began to ride their bikes in that manner. This grew into a sport and in '77 American Bicycle Association (ABA) was organized as a national sanctioning body for the growing sport. So it would not be uncommon for those bikes to be seen in 79.
When the bus overturns, the window is obviously breakaway candied glass, which breaks into large pieces and easily falls out of the frame (at around 1h 23 mins). Real vehicles use tempered safety glass, which shatters into tiny pea sized pieces but retains its general form. Although tempered glass was patented in 1900, it did not become federal mandate until 1977. Vehicles manufactured before that time may have in fact not had the same safety features known today. In addition, the relative cost of tempered glass instead of candied glass is prohibitive when shooting multiple takes of a movie. Even with its $50M budget, one has to assume that some lifelike replicas had to be made, rather than destroying actual vehicles and houses.
Although middle schools were not widespread until the 1980s, the National Middle School Association (now the Association for Middle Level Education) was founded in 1973, and is headquartered in Ohio, so the town in the movie could definitely have had a middle school in 1979.
During the fire scene there are several shots of rolling hills with horses (and other livestock) grazing in pastures (at 1:09:02 and 1:13:14). Horses, in particular, have a mortal fear of fire. With the impending fire, horses would have been running amok, neighing, and jumping fence rows to seek safety, not casually grazing.
When the Air Force weapons begin to misfire (and the kids are running through the chaos), an airman is seen yelling into a radio about the situation (at around 1h 26 mins). The radio he is yelling into is clearly not plugged in to his pack. This is obvious due to the open cannon plug on the bottom of it.
The goof items below may give away important plot points.
When the four boys are locked in the back of the military vehicle being attacked, Martin is seen clearly screaming and frantic. The very next shot shows him listening to the other talk in a relative calm way.
After the children are apprehended in the school, they are put aboard an Air Force bus. When the camera pans away from the bus and down to Donny in his car, the chain towing the bus is visible in the lower right corner of the screen (at around 1h 18 mins).
At the end of the film many of the soldiers guns are shown to be pulled out of their hands with considerable force. Moments later in the scene, most if not all the soldiers have guns back in their hands. However this may be not a mistake, since the alien appears to draw the objects to his construction selectively: Only one car gets picked up in the air from the parking lot, and by the end of the scene Joe's locket is the only object that gets drawn to the ship.