When John is talking to Frank in the past (1969) he refers to the murderer as being a "Serial" as in Serial Killer. Frank immediately understands what a "Serial" is but the term Serial Killer was not used until the 1970s. The term (serial killer) is widely believed to have been coined either by FBI agent Robert Ressler or by Dr. Robert D. Keppel.
The radios the firefighters use at the warehouse came into widespread use in the early 1980s. In 1969, they would have been carrying around radios that looked like a brick without an external microphone.
(at around 1h 35 mins) In 1969 Frank Sullivan is in the serial killer's apartment; he watches the serial killer remove a cigar box from a space in a closet, a moment later Frank retrieves the cigar box and we see that the cigar box is a Montecristo box which was not made until the late 1990s.
The 1969 world map does, correctly, indicate the Soviet Republics as distinct "countries" good for various radio awards. However, it looks like Rio de Oro (which in 1976 became Western Sahara after the Spanish left) is the same color as Morocco, which has occupied Western Sahara since soon after. In 1969, Rio de Oro would have had its own color.
In their first conversation via amateur radio, Frank tells John that he can't "broadcast" without a license. No ham would refer to two-way communication as a broadcast. The FCC rules that you have to learn in order to get your license specify that you cannot "broadcast."
(at around 1h 45 mins) In the final scene of the ball game, Frank Sullivan clearly hits a "softball" and it lands in the headlight of Gordo's car. The ball Gordo pulls from the headlight is clearly a baseball.
In the final scene of the ball game, the first view of the Mercedes' front license plate is a series of letters and numbers - like a traditional NY license plate. In the following close-up view, it is clearly '1 YAHOO'.
(at around 15 mins) When John is looking at the scrapbook, he turns to a page with a pasted-in newspaper clip that says "Amazin's Take Game 2." In the next shot, the clip isn't pasted on but folded in half and John has to open it to see it.
(at around 1h 5 mins) When 'Future John' comes up with a new suspect for the killings, Darryl, he shows his picture to Satch in the yearbook. The opposite page has writing on it. However, when shown again minutes later, the photo of 'Darryl' is opposite a page of photos of students.
(at around 1h 30 mins) When Frank is at the police station and uses the aerosol can and lighter to start the fire sprinklers, he directs the flame at the smoke detector which would have only set off the audible alarm, not the sprinklers. Additionally, only the fire sprinkler head(s) that reach a certain temperature due to fire/heat in the area will activate and begin spraying water... not the entire system.
Young John Sullivan is right handed when he plays catch with his father. Old John Sullivan is later seen writing with his left hand. It's not unheard of for people to use a different hand for writing and throwing.
In 2000, when John, Satch, and Gordo are sitting in the restaurant reminiscing about Frank, Satch reminds John that his father died 10 years earlier from cancer as a result of smoking. Not only is John smoking at that moment, but later that night at the radio table he continues to smoke as he is talking to his father Frank in the past. As John is smoking, he adamantly tells Frank that he will die from cancer as a result of smoking. It's a bit hypocritical for John to tell his dad to stop smoking while he himself is puffing away.
The goof items below may give away important plot points.
(at around 50 mins) When Julia saves Shepherd from dying, she warns the physician about administering Benadryl and benazepril together. Benazepril was not given FDA approval until 1991 and therefore could not have been used by the hospital. Furthermore, there is no fatal interaction between Benadryl and benazepril.
Just before Frank gets arrested we see that he is watching Game 5 of the 1969 World Series. Singer/actress Pearl Bailey sang the National Anthem for that game and while do indeed see her on the television we do not hear her rendition. What we do hear is a horn arrangement of the anthem.
In the ending John was able to talk to Frank on the radio while he was doing something else, but with the way that normal amateur radio equipment is most commonly used he would have to keep the mic button pressed while speaking in order to transmit. However, an earlier scene establishes that part of the seemingly nonstandard operation of this radio is that the talk button need not be pressed. In reality, most amateur radio transmitters and transceivers of this type have a feature most commonly known as "VOX", which is an abbreviation for "Voice Operated Xmit (Transmit)", and allows transmission without the need to push the button.
When Frank, in 1969, is saving the girl in the Buxton fire and John, in 1999, is being flooded with memories, we see his mother in memories of him being teenager, and graduating the police academy. However, shortly after that scene we find out John's mother became a victim of the Nightingale murders. During the funeral we see that John is still very young when his mother was killed, so she would not have been around during his teenage years nor see him graduate from the police academy.