In the beginning of the movie, you can clearly see both balls go through the tube at the same time.
But, in the scene where John finds he is going to commit his murder, he walks into the room and one of the balls comes down, he scrubs the image and sees his face. Just after that, the other ball comes down and sees his name on it.
In Anderton's case, the identification occurs late, because the Precogs have, as of yet, not seen the image of John Anderton, so they cannot identify him. If you think of John watching the scene live, as the Precogs are predicting it, you will understand why the ball is delayed.
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