Continuity mistake: They keep showing the aerial view of one type of jet (such as Gulfstream 100 with its rectangular windows), but the inside shots are always in a "heavy" Gulfstream (such as Gulfstream IV or Gulfstream V, which have round windows). Also, the tail numbers (the N-numbers written on the engine) are always different in the aerial views. Some end with GV, some with SP etc.
Continuity mistake: When Gideon is talking when we first see him, the pictures behind him on the screen change instantly. It should be noted, that when we see the pictures change, there is the audible click and visual delay of a slide change which is not present on the instant change. (00:04:20)Ssiscool
Continuity mistake: At the start we see Spencer explaining to Gideon about the Boston Strangler, his orange shirt is untucked at the front with both sides of his shirt being visible. Camera shows Gideon walking off and now only 1 side of his shirt is untucked. (00:05:10)Ssiscool
Character mistake: Morgan asks Reid what track on the CD (Some Kind of Monster by Metallica) would help an insomniac sleep. Reid replies with "Enter Sandman" however, Some Kind of Monster doesn't feature the song Enter Sandman. (00:29:00)Ssiscool
Other mistake: Just after they were finished explaining that you can not tilt the mystery package because it would instantly blow up, they show the little girl at her doorstep holding the package completely tilted. Just when you think that means that this package now doesn't fall into the "explosive" category, they then show the mom freaking out, and later police making the girl hold the package in a completely different (non-tilted) position.
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 threw the child at Gideon, who fortunately caught him.