When Arthur is taking August to Wizard's for the first time, August is holding a can of soda in one hand, while Arthur is holding the pizza box. In the next shot, August is holding the pizza box, but no can and there would have been no way he would have disposed of it or taken the box from Arthur.
The license plates on the cars in NYC have the design of the current New York License plates. However, the part of the movie where Louis is chasing Lyla is supposed to take place 11 yrs prior to present day, being 1995-1996. Those license plates were not issued until many years later, probably 1999-2000.
When August was playing the piano at the church for the first time, he played a note, then two more to the left, and got a higher tone when he should have gotten a lower one. He then plays two notes to the right of the original pitch, and gets lower tones. This is exactly reversed from what should have occurred.
When Evan (August) is at the pipe organ in the church, at one point he is seen from behind. At this time, one hears some high notes descending in a scale-like pattern. To play this, August's right hand would have to be far to the right of the organ keyboard. It is not; it is close to his body, near the middle of the keyboard(s).
When Louis's brother takes the picture of Louis and Lyla on the rooftop couch, Lyla has her hand on Louis's arm across her chest, and he has his other hand on top of hers. Whenever you see the photo for the rest of the movie, however, his hand isn't over hers.
Robin Williams' character "Wizard", depending on the scene, appears to have two different hair colors. In some scenes, he appears to have red hair and sideburns and in other scenes a brownish auburn color.
One of the signs that the Wizard tears down (we see a close-up of it as he rips it off a pole) says Evan Grady - Missing, not Evan Taylor. This is not long before the police raid the place where the Wizard and all the kids stay.
A policeman chases August out of the former Fillmore East and up the stairs of an elevated subway station, but there are no elevated subway stations in lower Manhattan, where the Fillmore East was located.
During Louis' unsuccessful trip to Chicago to try to locate Lyla, he's briefly questioned by two police officers in a squad car outside her apartment. Chicago police cars have blue roof lights, but this one has a red light bar on top, and appears to be a New York squad car, filmed from a tight angle to obscure its actual markings.
In addition to the already-reported lack of an El train on Second Ave. in front of the site of the former Fillmore East (there once was a 2nd Ave El but it was torn down in the 1940's), the film portrays the Second Avenue facing front of the former Fillmore East as a dilapidated version of its former self, still sporting the old Fillmore East's marquee & still labeled as such. In fact, the Fillmore East's former entrance/lobby area is and for quite some time has been a redeveloped bank branch.
Because of his despair for losing Lyla, Louis does not play his guitar for eleven years. When he decides to start playing again, he takes out his guitar and plucks every string. His guitar is perfectly tuned, and this makes no sense. If you go so long as a month without touching your instrument, it will go severely out of tune. And he didn't tune it off camera before he did this, as we see him open the dusty case for the first time in a decade and play the strings immediately afterward.
Evan (August) goes missing in February of 2006 (according to the missing poster), and mentions to Louis that he has been playing for six months. This isn't clear whether he means only his time at Juilliard, or the time up to and including his time at the school. It is possible to have him missing in February, be on the streets, study at Juilliard, and have the spring concert in Central Park, depending on how long August was on the streets. This is borne out by a scene (while August was in Julliard) of Louis sitting in the park in the snow; August must have entered Juilliard in the fall, and had his concert in the spring.
When Evan/August is in the church, he starts writing down his composition, using the proper symbols for each type of note, even complex ones that only experienced musicians would know, but there is no explanation of how or where he would have learned these notes.
When the hospital is calling to inform Lyla about her father, her phone rings and then you start hearing the message being left with no time for the other person to hear the outgoing message that we hear later in the movie.