Any alternative characters to clarify the difference, as the English words and their Greek roots have different meaning. Thanks!
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.
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.
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
sympathy translates as "together feeling", so 同情， 同情心 are good. sympathy has an antonym: 冷淡, but empathy does not. Empathy is an ability.
empathy is a relatively new word, the English version of German '(sich) einfühlen.' = 'in feel, feel your way in'
Originally in relation to art, empathy is: "A term from a theory of art appreciation that maintains appreciation depends on the viewer's ability to project his personality into the viewed object."
empathy: emotional transfer 感情移入
In modern parlance, empathy and sympathy are confused.
To me, 'empathy' is the ability to understand others' feelings and problems. So, the correct Chinese is 同感力 and I also see some translate it as 同理心, 共情 and etc.
'Sympathy' is the feeling of being sorry for someone who is in a bad situation. So, 同情 or 同情心 can fit for this sense.
I think 'empathy' and 'sympathy' have some overlap in meaning because if you have great empathy with people, it also suggests that it's easy for you to feel sympathy for them.