Confusion of "empathy" and "sympathy" abounds since different people, with different vantage points, use the terms differently. In scientific studies within Psychology, empathy is usually defined with perspective taking. Various kinds of empathy are distinguished with sub-scales, such as cognitive empathy (thinking what others think by considering their perspective), affective empathy (feeling what others feel), or internalizing from abstractions (e.g., feeling like you're a character in a novel). From this perspective, I've heard people say another word for "affective empathy" is "sympathy" (but generally don't use the word "sympathy"). The empathy-altruism hypothesis is the recognition that empathy is among the best predictors of altruistic behavior, especially when other predictors are absent (like the absence of a reciprocity norm).
Even within Psychology, perspectives vary! Helping professionals sometimes distinguish empathy and sympathy by how wrapped up we get in the feelings of another. In this context, it's usually healthier if you minimize empathy, because it blurs boundaries. Instead have sympathy, where you maintain boundaries but can be helpful. So ironically, some helping professional see sympathy, not empathy, as the key to effect altruism.
Common English usage of "sympathy" is something you have and share with another person experiencing misfortunate (e.g., "my sympathies") whereas "empathy" describes a person who is good at taking the perspective of others (e.g., "she is so empathic."). I've also heard people in common English usage describe empathy as something you can only have if you've been in the same situation yourself (something Psychology normally doesn't).
In short, I feel it's totally understandable why this is so confusing. You might say ...
I empathize. :-)
... and sympathize with your effort to figure this out, Maurice. :-)
I agree with everything Dr. Grobman has said above. A neat (but over-simplified) distinction has been made between sympathy as "feeling with" and empathy as "feeling into" - to emphasise that sympathy is primarily affective, whereas empathy is both cognitive and affective.
Empathy is the ability to understand things from someone else's point of view, to put yourself in someone else's shoes, the ability to experience another person's feelings. Empathy leads to a better understanding but not necessarily compassion. A good actor who is playing the role of the villain needs to have empathy for his character to act it out good but he may not feel compassion for his character. So empathy equates to understanding things from someone else's viewpoint.
Sympathy means to have compassion for others' misfortune, but without necessarily seeing their troubles from their viewpoint. When there is sympathy without empathy, there is more compassion but less understanding. A person may feel pity for others without really experiencing their feelings. For example, a person may feel sympathy for all differently-abled people because it is common in his culture to feel compassion for differently-abled but he does not understand differently-abled people's real problems and concerns from their viewpoint.
Sympathy= feelings of pity and sorrow for someone else's misfortune. Eg."they had great sympathy for the flood victims". Empathy = the ability to understand and share the feelings of another. Eg. Friends/ relatives departure moments.