Factual error: When August sits down at the piano in the church and begins to play, he plays a passage of ascending notes (his hand is actually on camera so you see it move to the right, up the piano). However, the audio that you actually hear is a passage of descending notes.
Audio problem: When Evan plays the guitar the first time, you can hear some chords and notes that aren't actually being played at all.
Continuity mistake: When Evan is playing the guitar for first time in front of Wizard and the others, Wizard is wearing a white shirt, but when Evan and Wizard are talking looking at the stars, Wizard's clothes are different.
Deliberate mistake: When Evan/August first meets Arthur at the park, after playing, you see Arthur leaving his spot without taking the cable and (presumably) the amp as well.
Plot hole: When Mr. Jeffries finds out that Evan is named August Rush too, he goes to the Central Park concert and sees Evan onstage, however he couldn't know that Evan was having a concert, or even that Evan was at that place at that time.