Empathy and Sympathy use same characters in Chinese 同情?
From Wiki


  Empathy is the capacity to understand or feel what another person is experiencing from within their frame of reference, that is, the capacity to place oneself in another's position. 


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  Sympathy is the perception, understanding, and reaction to the distress or need of another life form. This empathic concern is driven by a switch in viewpoint, from a personal perspective to the perspective of another group or individual who is in need


Base on the definitions, translate 'sympathy' to 同情 is correct, but the meaning of 'empathy' should be 體諒

"同情" means "pity" or "sympathy"

"體諒" means "considerate/ understanding" , notice the components of "體諒" are 體(experience) + 諒(understand)-- To 體諒 means "you put yourself in another's position and understand how the other feels"

"體諒" is not the same as "同情" 


"同情" someone, means you are pitying, feeling sorry for that someone
When you say you have empathy (體諒) for someone, it doesn't necessary mean you feel sorry for that someone. 


For example:

我能[體諒]美國一部分白人要保自己的特權階級地位而投川普的票，但我並[不同情]他們

I can [have empathy] for those white Americans who wanted to protect their privileged class status and voted for Trump, but I [have no sympathy] for them.

Of course if you have empathy for someone, you are much more likely to have sympathy toward that someone as well