The city Arzamas-16 was renamed Kremlyov in 1991 and since 1995 its official name is Sarov. It is possible that officers use the old name out of habit but that doesn't explain mistake in the forced subtitles.
When Ryan tries to summon help after his helicopter crashes, he reaches into the cockpit and turns a knob on the radio transmitter. The film has been flipped and the word "frequency" under the knob is clearly spelled backwards. This would also explain why Ryan reaches through the cockpit window with his right arm, but it's his left hand that adjusts the radio frequency.
There are at least two different "Tail Numbers" shown for the President's aircraft - a four digit (42xx) tail number is used when the aircraft is on the ground, and later a five digit (7xxxx) number is used when the plane is flying.
As Ryan's helicopter crashes after being hit by shockwave, you can see one of the crewmen almost get thrown clear off the chopper's right front window. When Ryan climbs out of the destroyed aircraft, the crewman is nowhere to be seen; later on, when Ryan reaches for the radio, the crewman's body is farther inside the aircraft.
When Mason is loading the crate containing the bomb into his truck, the forklift pallet is clearly still attached to the bottom of the crate. When Mason reaches the stadium and unloads the crate, the forklift pallet is missing.
After the helicopter crash, Ryan uses the pilot headset. The mic on the headset changes from green (David Clark H10-40 style) to black (H10-30 style) indicating a different headset was used on that shot.
An early shot in Moscow shows a group of 3 cars driving towards St Basil's Cathedral, with the Kremlin to the left. This is a flipped shot, as the Kremlin should be on the right when driving towards the front of St Basil's Cathedral.
When the small jet lands back in the USA after the visit to see the Russian President, the engines are just shutting down, the door opens, and you can see a cable attached to the plane. There was not enough time for ground crews to have done this.
The USS John C. Stennis is a naval carrier home ported in Washington state. It would not be operating in the North Sea to allow the Russian Air Force to attack it. It was built on the East Coast but did its world cruise in the late 90s. Since then it has operated out of the Pacific Ocean and mainly the Middle East.
As the president's motorcade is shown driving towards the Baltimore Forum (located in the inner harbor) we see a helicopter shot of the motorcade headed out of town on I-395 and turning east to go directly through the Ft. McHenry Tunnel, away from the inner harbor.
The people digging out the Israeli bomb look like nomad Bedouins in a tent camp. In fact, at least on the Israeli side of the Golan, there are no Bedouin inhabitants. The Golan Heights are inhabited only by Druz and Israeli Jews.
When the "Baltimore Forum" is shown in initial aerial shot, it is located on O'Donnell Street (just off I-95), therefore the Presidential motor adding is traveling in the correct direction to that location. Also, it is shown as a domed stadium, thus the lighting of the footage of the game is appropriate. The aerial shot of Baltimore shows M&T Bank Stadium (as well as Oriole Park at Camden a Yards), technically not the same stadium as where the game is being played as established by the earlier shot of the "Baltimore Forum".
The effects of the chemical attack on Chechnya are described as resembling "acute late-stage cerebral palsy." Cerebral palsy is a birth injury, not a progressive disease, and therefore does not have stages.
The Presidential motorcade is not only short an ambulance, it is also missing a large portion of police presence, at least 3 times less than there would be there in reality. Also, if not directly above, there should be a chopper near by.
When the President and his aide give the order for a nuclear attack, it is said that about one minute will pass before the missiles start. In reality the Launch Control Centers alone need between three and five minutes to order the launches.
When the nuclear bomb detonates, its shock wave is strong
enough to flip the presidential limousine in the outskirts of the city over. It would have to have a yield of at least 1 megaton to do this, but news broadcasts describe the explosion as being smaller than that of the Hiroshima bomb. The Hiroshima bomb had a yield of 15 kilotons.
The three primary effects of nuclear detonation, flash, heat and blast, are confused. In the blast scene from inside the hospital, the shock wave and flash of light arrive together. The only way for this to happen is if the hospital was right next to the stadium, in which case the hospital would be obliterated. Furthermore the heat from the blast appears to have no effect. Everyone else, further from the explosion, would first be blinded/burned by the light/heat before the shock wave would hit them, as the shock wave travels only at about the speed of sound. Also on the TV the blast in the stadium is seen the fraction of a second before the blast hits the hospital. Even with a direct link to the camera in the stadium the picture of the blast couldn't arrive earlier than the flash of the light from the bomb itself.
When President Nemerov first meets Jack Ryan, he is having his portrait taken with a Sinar 4x5 view camera. We correctly see the President's image upside down in the camera, however the photographer continues to take pictures with an empty camera. In a 4x5 view camera film is loaded with individual sheets placed in front of the glass viewscreen. Seeing the President's image in the camera would be impossible if the camera were actually loaded. Also the sound effect is that of an SLR medium-format camera which uses a focal-plane shutter and flipping mirror. The Sinar uses neither. Its shutter is actually inside the lens and makes very little noise.
The nuclear suitcase is wrongly described. In reality the suitcase contains red plastic cards with codes, that have to be transmitted to the nuclear forces. The identification codes on the cards that the President and his aide have to carry at all times with them are not enough.
Dressler's servant says "Bist du verrückt..." (are you crazy...). A person living in a German-speaking nation would know to address another that is not his/her personal friend with "Sie", the polite form of "du" (you). The error is more or less confirmed as Dressler himself moments later announces his servant as "Herr Haft" (Mr. Haft), using Mr. followed by a surname, the correct, polite way to address someone who is not an acquaintance in German.
When Clark is using the spray-can chemical agent to instantly cut through the chain-link steel fence, he is not wearing any protective covering (mask, shield, etc.) over his face. This material is obviously very powerful stuff. Had a sudden breeze, or some splash-back directed any of it onto his skin, or into his eyes or nostrils, he would have been immediately incapacitated.
After the bomb explodes, the President's limo is flipped upside down, and a team of Marines is sent to rescue him. As the CH-53 lands, the Marines run out of the back of the helo in all directions (both left and right). This would not be possible on an actual CH-53, as the CH-53 has a low tail rotor mounted on the pilot's side of the aircraft.
According to the scoreboard, Florida is playing Chicago in Baltimore. Chicago is in the "home team" spot on the scoreboard. Theoretically, the Super Bowl could be in Baltimore and feature two out-of-town teams, but the weather was summer and not February.
The vending machine with the bomb has a length of twisted red-and-black wiring showing at the front, although there is no reason why this should be so. Clark finds some pieces of this wiring in the factory in the Ukraine, but all this means is that some electrical equipment was being worked on; it certainly doesn't prove it was a nuclear bomb.
When the President arrives at the football game it is broad daylight. When the President leaves the game and the subsequent explosion occurs it is also broad daylight. The scenes of the actual football game however, are either at night or inside a lighted, dome stadium. A satellite view of the city just prior to the explosion shows only an open air stadium, (which appears to be empty) again in broad daylight. The dark, lighted scenes of the football game do not match any of these other accompanying daylight scenes.
The goof items below may give away important plot points.
The Russian commander orders a strike on an American aircraft carrier, which is shown with virtually no defenses. Aircraft carriers travel in groups, generally as a Carrier Battle Group or a Carrier Strike Group, consisting of guided missile cruisers, destroyers, frigates and submarines, but never alone, and would be protected at the perimeter by these other ships. Thus, the inbound anti-ship missiles fired by the Russian aircraft would have been surgically removed from the air by the other ships' Phalanx CIWS (close-in weapons system), and it would be highly improbable that even one anti-ship missile would have gotten through.