A photocopy is shown of the dead Soviet spy Rob's driver's license. The license gives his year of birth as 1966, which if correct would make him rather young to drive a car in 1981.
When photographs of the Soviet operative are shown, Beeman comments that he has "a gaping stab wound in his kidney". The wound shown is near his navel. Not only it would not touch his kidneys, but, considering the very short blade of the knife shown in the pilot, it's not even lethal.
When Robert's wife and baby get picked up, it shows a sign for 117th St. There is no 117th St. in Philadelphia.
While trying to dodge FBI agents on the streets of Philadelphia, Greg is spotted standing behind a mailbox displaying the new modernized eagle USPS logo rather than the previous version, which would have been in use in 1981.
When Phillip and Elizabeth are questioning Joyce, the baby she holds switches between a live actor and a doll in between shots.
(At around 10:00) there is a calm but suspenseful outdoor scene, where, in a span of one minute, you see three 1990's and newer vehicles in the background: a 2010 Chevrolet Silverado, a 1997 Volkswagen Cabrio and a 2010 Nissan Juke.
When Phillip is showing the briefcase full of money there are Series 1966 $100 bills. After the money is thrown in the air, Series 2006 $100 bills are shown on the floor. Those bills didn't come out for around 20 years after the time in which the show is taking place.
When Philip drops the suitcase of cash on the floor, several of the one hundred dollar bills have the oversize, off-center portrait of Benjamin Franklin. "The Americans" is set in in the early 1980s, but this version of the $100 bill was not introduced until 1996.
Gregory is helping Elizabeth by scouting out a person of interest, and several automobiles from the modern era are seen in the distance.
Beeman reads the name on the driver's license as "Robert Owen McKenzie" But there is no middle name on the License.