Let’s look at the nuances of these terms:
Empathy, Sympathy and Compassion are not mutually exclusive and can occur at the same time. People who lack the capacity of Empathy, Sympathy and Compassion consistently may be classified as narcissistic, sociopathic or psychopathic in rare cases.
The words used above are culturally dependent making them more of a description than a scientific definition. To dig more deeply into the physical brain response, check out this white paper by Dr. Ron Bonnstetter.
Here’s another interesting quote from an article by Dr. David R Hamilton.
You can actually see the difference in the brain. Empathy lights up the insula, which is an area of the brain that connects the flow of information between the front of the brain and the emotional areas, whereas Compassion sees many of the same areas lit but with the addition of the prefrontal cortex (left side), which is the area above the eyes that is involved in decision making.