I haven't given much thought to this subject for many years. With the help of some Buddhists friends I came to realize that compassion and sympathy are distinct.
Sympathy is about identifying with another being; as one would say ‘I can imagine being in their shoes’. That is sympathy.
Compassion on the other hand deals with beings for whom you cannot have a personal identity such as an ant, an oyster, a mosquito or even a log.
What has become part of this discussion most recently is the observation that compassion may leave one completely unable to act. Compassion requires that one feels the same emotion for a being that is attacked as one does for the being doing the attacking.
Compassion therefore restrains any form of action. However humans invariably will act when they see what appears to be an injustice. Therefore compassion, as understood by humans in our daily activity probably doesn’t exist. We probably use sympathy for most actions that appear benevolent. Actions require a judgement based on the context. When I see a cat attacking and then eating a Morning Dove compassion requires that I see both beings as equals.
If possible I will invariably try to stop the cat from eating the Dove. In this context I can ex post explain my behavior but the reality is I am NOT truly exercising compassion when I value the bird’s life over the cats interest, whatever it might be.