Det. Cross may be highly intelligent, however, his actions as a police officer are lacking. When he encountered an armed highly skilled assassin that had killed multiple armed opponents, he did not shoot him. This of course would have negated the rest of the film.
In the final scenes of the movie:
SPOILER: When Picasso falls in the car at the end of the movie, his body slips down and his head is placed between the hood and the windshield. When they are examining him, his head is in the middle of the windshield.
An RPG launched from a moving train will fly laterally in the direction of the train's movement, as well as towards the target. Not directly straight to the target as if it was fired from a stationary position as shown in the film.
When Detective Monica Ashe is first giving details about Fan Yau Lee to Detectives Thomas Kane and Alex Cross just after Fan Yau Lee's murder she says that Fan Yau Lee received an MBA from a university in Shanghai and was then awarded a "Rhodes Scholarship". Rhodes Scholarships are awarded to deserving individuals from fourteen specified geographic constituencies. Mostly, these geographic constituencies are, present or former commonwealth countries of the United Kingdom. China, the country where the city of Shanghai is located, though is NOT one of them.
The end credits incorrectly attribute the hymn tune "Faith of Our Fathers", sung at Maria Cross's funeral, to Colleen Coil. This was merely an arrangement of the 1849 work by Frederick William Faber, with music from 1864.
When Picasso is sneaking into Erich Nunemacher's building, the screws are shown turning clockwise to loosen. Screws (with some very rare exceptions) don't work that way. Screws turn counter-clockwise to loosen.
Several routine police procedures are absent. Since the killer had called the cell phone of Det. Cross, all his phones would be monitored. The funeral would have surveillance. High profile threats would involve a SWAT team, not just 3 detectives.
Towards the end of the film, when two cars collide, Alex Cross' airbag inflates. He then cuts the airbag away in order to leave his car. Airbags immediately deflates after initial inflation, so there would be no need to cut it away.