Continuity mistake: In the scene where Avner shoots the first Palestinian, the dead man falls forward onto his groceries, but the next time the body is seen, he is lying on his back.
Factual error: In the final scene with the New York skyline in the background, the new glass/stainless steel Manhattan complex at 731 Lexington Avenue is seen; this building is just being finished now in early '06. The building stands where Alexander's dept store once was. (On a related note, it is a nice touch on Spielberg's part to overlay the Twin Towers onto the skyline in that same scene.)
Factual error: In the scene towards the end where Avner is speaking with his mother, there is a central-air-conditioning-type vent in the wall. It seems to me unlikely that an Israeli house/apartment in the 1970s would have had such a system. Air conditioning units were not very common and what would normally have been found was a single-piece unit installed through a wall.
Factual error: In the background at Avner's debriefing is a water cooler. Israel's two earliest mineral water companies (Mei Eden and Neviot) weren't founded until the 1980s, so it is unlikely that there would have been such a water cooler there (assuming that the debriefing was taking place in the mid/late 1970s).
Continuity mistake: In Paris, when they call Mahmoud's number the first time, the first digit dialed is a 4. The second time, they begin with a 7.
Continuity mistake: Near the end when Avner and Steve are attempting to assassinate Salameh, they shoot a man while he is standing easily 20 feet away from the pool, the next shot is of him falling in the pool.
Revealing mistake: During a flashback to the original hostage-taking, we see one hostage get shot and the camera pans to the wall behind him being riddled with bullets. Only problem is that the paper on the wall is bending outwards, caused by the squibs behind it, rather than being pushed inwards by the path of the bullets.
Continuity mistake: When Ephraim and Avner are walking along the boardwalk in sunny weather and Ephraim is proposing the "job offer", in the side shot they walk past two pedestrians, one of whom is leaning on the railing. In the next shot facing them, as they are still walking forward (now towards viewers), they are walking past the same couple again.
Factual error: The 5 German 'snipers' at the Munich airport are shown using G-3 rifles with scopes on them. None of the 5 snipers had scopes on their rifles. In fact, they didn't even have semi-automatic rifles. They used bolt rifles with iron sights. Their poor equipment contributed significantly to the failure of the rescue operation and prompted H&K to develop the PSG-1 sniper rifle.
Factual error: In the scene about half an hour into the film when they're walking through a street, (it's supposed to be Rome but its shot in Budapest) you can see parking meters in the background that are activated by SMS; obviously such technology did not exist in 1973.
Factual error: When Avner is in London to try to kill Ali Hassan Salameh, vehicles are seen driving over a raised junction - a traffic calming measure, which wasn't introduced until more than 20 years later than 1973.