What is the difference between heroin and ordinary medical morphine?

"Morphine" as it is commonly referred to, is morphine sulfate. Heroin is diacetyl morphine. That is, heroin is simply morphine with an acetyl molecule attached.

In terms of effects, they are exactly the same -- and medically interchangeable -- except for dosage. In fact, they are both converted to the same form of morphine when they get into the body. 

The only significant difference between them is that the acetyl molecule allows heroin to cross the blood-brain barrier more quickly than ordinary morphine. The result is that, in terms of dosage, heroin is about three times stronger. That is, one grain of heroin equals about three grains of morphine. Otherwise, they are identical and there is no significant difference that would justify heroin being completely illegal (Schedule I) while morphine is used routinely in medicine. (Schedule II) They can be used interchangeably as long as medical personnel follow standard practices of medical care.

At this point a sensible person might ask why heroin is illegal while morphine is not. The answer is the same answer about the origins of almost any of our drug laws -- plain old legislative ignorance of the subject. 

You can find a good description of how it happened in When was heroin outlawed, and why?