Other mistake: When Glen Holland hears that John Lennon was killed, he is sitting at his piano and outside the window it is sunny and leaves are on the trees. As he walks home, it is overcast, snow is on the ground, and the leaves are off the trees. Since Lennon was killed in December of 1980, the latter "weather" would be correct.
Revealing mistake: Anytime the camera is on Richard Dreyfuss' hands while he's playing the piano, it is clear that he is not actually playing; his hands are frequently in the wrong part of the keyboard for the sound being produced. At other times, when his hands are moving down the keyboard, the pitch we hear is going up, or vice-versa.