EMH was designed to be a doctor, not a janitor. I bet a lot of the work behind EMH was in proving mathematically that it would not harm patients (while at the same time, understanding that some temporary harm might be needed to save a patient's life). We see how core this is to an EMH's identity when in Voyager, the doctor goes crazy after not being able to save a crew member.

Meanwhile, Moriarty had no safeguards. That's part of why he was so dangerous to the crew of the Enterprise.