compassion vs fellow feeling

compassion and fellow feeling both are nouns.

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

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

compassion: a deep awareness of and sympathy for another's suffering
fellow feeling: sharing the feelings of others (especially feelings of sorrow or anguish)

Other hypernyms of compassion include sympathy.

compassion (noun)	fellow feeling (noun)
the humane quality of understanding the suffering of others and wanting to do something about it	sharing the feelings of others (especially feelings of sorrow or anguish)
a deep awareness of and sympathy for another's suffering