empathy vs fellow feeling

empathy and fellow feeling both are nouns.

Word	Noun	Adjective	Verb	Adverb
empathy	Yes	No	No	No
fellow feeling	Yes	No	No	No

As nouns, fellow feeling is a hypernym of empathy; that is, fellow feeling is a word with a broader meaning than empathy:

empathy: understanding and entering into another's feelings
fellow feeling: sharing the feelings of others (especially feelings of sorrow or anguish)

Other hypernyms of empathy include sympathy.

empathy (noun)	fellow feeling (noun)
understanding and entering into another's feelings	sharing the feelings of others (especially feelings of sorrow or anguish)