Character mistake: In the initial meeting at FBI headquarters, the agents mention the Zodiac killer. Gideon says about him "Yeah, and he killed for 30 years without being caught." In actuality, the Zodiac's known victims were killed between 1966 and 1970. In letters to the public, Zodiac claimed responsibility for several other murders (most of which were solved without being attributed to Zodiac), but no later than 1974. Gideon's statement was off by at least 22 years.
Factual error: When the professor gets killed, a kid is shown trying to drink out of the water fountain, but the water is turned off. Gideon figures out that there will be no water to the sprinklers, so a fire is soon going to be set. In real life, fire sprinklers are not connected to the main water supply. Sprinklers are on a separate pump.
Factual error: In this episode Gideon mentions Andrei Chikatilo, and says that he lived in Ukraine (and that this is why he remained uncaught for so long). While Chikatilo was born in Ukraine, he lived in Russia for most of his life, and all of his murders happened here. An expert like Gideon should not make this kind of mistake.
Continuity mistake: At the beginning, a St Louis couple has reported their son missing. At the FBI briefing, the agents are told that the boy's mother tucked him in at 10:00pm. A few minutes later, we hear the radio call over footage of police at the home, and the dispatcher says "Last seen at 2100." That's 9:00pm, not 10:00pm.
Character mistake: Derek Morgan says that he discovered the body of a dead boy in 1991 when he was 15 years old. Later on, you see Garcia searching the computer for personal info on Derek Morgan. According to the computer; Derek Morgan was born in 1973 which would make him 18 years old in 1991, not 15.
Continuity mistake: In the scene just after Gideon talks to Cherish's mother we see him start to walk toward the rest of the team. In the wide shot we see Reid standing with his hands in his pockets and jacket open, we then see him close his jacket and cross his arms as Gideon gets closer, but in the very next shot (the close up on Reid) we see his jacket open and his arms at his side once more.
Continuity mistake: In the scene where you first watch the killer drain a victim's blood, the upper part of the glass jar is coated with blood, but when he puts it in the fridge this part of the jar is clear. On the other hand some of the jars in the fridge are still fully coated with blood, which shouldn't be as the blood had the time to gather at the bottom.
Plot hole: In the scene where The Reaper/George Foyet calls Haley from Marshall Kassmeyer's home he identifies himself as Victor Collins from the Marshall Service. Later in the episode Hotch calls Haley and tells her he needs Jack working the case. Jack comes to the phone and asks him "Is George a bad guy?" Until Hotch called Haley she didn't know Foyet was not a US Marshall named Victor Collins.
Continuity mistake: Before the BAU catches the serial killer Carl Arnold, they used a heat sensor and it shows five people sitting at a table. They said there were three adults and two kids, but when they break in the house, the mother and one kid were sitting at the table, the father was tied up nearby, and Carl Arnold and the other child were in the basement. They were found by Derek Morgan right before Carl Arnold killed the child.