Plutonium must be alloyed with another metal (usually gallium) in order to prevent forming allotropes which cause it to crack while cooling. Cracks in the pit would have significant impact in the weapon, and could result in a fizzle (non-nuclear explosion.)
During the process of making the sphere of plutonium Paul doesn't use any type of protection (filter) from airborne plutonium particles. There is a high probability of death when plutonium is inhaled. As a minimum it results in radiation sickness.
Paul shows Jenny a fistful of four leaf clovers that he had picked some time ago from the grounds of the lab. When revealed, they're crisp and freshly picked. If they were in his pocket for as many hours, they would have withered and been much darker in color.
The specific details of the construction of the bomb and all its component parts are incorrect (obviously).
Paul's mother is surprised when he gets up at 4:00 in the morning at the beginning of the movie. Later on, she tells Mathewson that Paul has been getting up this early - and earlier -regularly for years.
The security guard spends several minutes in the rain. When he goes into the lab to investigate he is dry and leaves no wet trail.
Every time Paul opens the glove box in Dr. Matheson's Mercedes he picks the lock with his nail file, but never locks it again. The glove box lock is simply set to lock automatically when closed.
A plutonium pit weighs at least 5.5 kg (~11 lbs), but Paul lifts it out of the tube with a magnet attached to what must be the steel shell he used for casting.
Paul's dropped rubber glove mysteriously appears on his hand as he reaches for his bag.
As the townsfolk drive to the lab at the end of the movie, as the guy on the scooter heads offroad, his scooter becomes a dirtbike.
When Paul picks up the toy car with the Plutonium he is kneeling on his knees in the dirt. There was just a very heavy rain storm, yet his pants are dry around the knees when he stands up.