Continuity 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.
Continuity mistake: In the scene where Mr. Holland is introducing himself to the class, he writes his name on the board. The "A" in his name switches between having a rounded top to a pointed top between shots.
Continuity mistake: At the end, when Mr. Holland is conducting his symphony, there is a closeup of Iris lifting her camera up to take a picture. The shot cuts away to show Mr. Holland. You can see Iris and Cole in the background on the lower right of the screen and they are holding hands with no camera in sight.