Deliberate "mistake": This error is found 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 (although you only see the cables on the ground and the amp is cut off).

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.

Other: Because of his despair for losing Lila, 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. As I am a guitar player myself, I know that 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 afterwards.