After the train wreck, Alice's car is dirty, covered in ash and debris When the kids get in the car to leave the train station, the car is clean.
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.
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 kids are watching the home movie it is clear they are in the projector's light but the cut to the film playing has no shadows on it (at 58:51 as Joe and Alice watch a film of young Joe).
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).
Hitting a pickup truck will not cause the massive train wreck depicted (it will just kick the pickup off the tracks), even if it it did, the railroad cars will not have the inertia to fly through the air.
When the sheriff is filling up his car at the gas station, he opens the fuel flap and sticks the nozzle straight in without removing the filler cap.
Dr. Woodward uses his truck to ram the train. Considering that the train was made up from several multi-ton train cars going about 50 mph, the truck should have been completely obliterated in the collision and the doctor killed on impact. The truck as seen by the kids, hits the train head on. After the crash, the kids find the truck with only the some of the right side of the truck demolished and Dr. Woodward injured but still alive.
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.
During the town hall meeting one guy says he had interference on his ham radio. Officer Lamb asks him what frequency he heard the interference on. The guy said it was on 13201, 13MHz is not in the amateur radio bands.
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.
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.
In the kids' mini film the doctor drops the syringe and the detective picks it up. In one shot, the syringe is one way in the detective's hand (yellow end by the heel of his hand). In the next it is reversed (yellow end by his thumb). However, the director of that mini-film was a school-kid who, whilst keen, may easily have overlooked a few things. It may even have been an homage to films of that genre.
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 $50 million budget, one has to assume that some lifelike replicas had to be made, rather than destroying actual vehicles and houses.
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. However, we know that the cubes initially morphed from being the hull of the spacecraft, and therefore can be certain of little about their shape-changing capabilities.
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.
Seeing the alien's claw marks in the walls of the caverns he created after initially tunneling down in the cemetery's storage building, one may wonder where the huge mountains of displaced dirt went.
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.
While the neighborhood used in filming may have looked the same to locals as it did in 1979, there were actually far more analog television antennas mounted to roofs still in use during that era.
When the train passes the Lillian Depot, the locomotive has a different horn sound.
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.
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.
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 almost certainly didn't exist in 1979. The 'R' in particular indicates it's probably Vera Sans from 2002. Another suggestion was that it was Ariel from 1982, but that does not have the appropriate level for the middle horizontal in the 'R', in particular relative to that of the 'E', even though the distinctive tail is a good match.
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.
When Joe is speaking to Alice on her front doorstep (from 30:01 to 30:06), modern vehicles can be seen driving on the street in the background.
time stamp 49:33, the map used shows Beavercreek as a city. At the time period of the movie, Beavercreek was not yet a city and was not named on any known maps of the time. Beavercreek was only a township and did not become a city until early 1980. It was a couple years later before it appeared on any maps.
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.
When father Louis Dainard wrecks his car and begins hearing daughter Elle Fanning (Alice Dainard) screaming, he starts calling her by the name of Elle.
While the blue name tapes and stripes were correct for the USAF in 1979, the rank on Colonel Nelec's fatigues should have the bottoms parallel to the collar (thus appearing to face downward, and not horizontal, when viewed from the front).
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.