In The Sixth Sense, how did the psychiatrist (Bruce Willis) fail his patient/ultimate killer?

As the plot goes, Willis is shot in his bedroom by a former patient who is angry at some mistake his doctor made while treating him. Later in the film, Willis is going over his tape recordings of patient sessions, listening closely for any hints as to what happened during his last appointment with this patient. The scene hints that Willis comes to some realization about what went wrong that set his patient on a downward path, but I have never been able to figure out what the big deal was. The scene suggests that Willis had to leave the room to attend to another patient with a seemingly more urgent problem, something that might upset/offend a patient, but this is a vague explanation given the "AHA" reaction of Willis listening to the recordings. Was the explanation any more specific than a doctor temporarily leaving one patient for another? Or did these tapes reveal something specific about the mental state/problem of Willis ultimate murderer when the doctor left the room?