care vs feel for

care is a noun but feel for is not a noun.

Word	Noun	Adjective	Verb	Adverb
care	Yes	No	Yes	No
feel for	No	No	Yes	No

As verbs, feel for is a hypernym of care; that is, feel for is a word with a broader meaning than care:

Other hypernyms of care include compassionate, condole with, pity, sympathize with.

care (noun)	feel for (noun)
activity involved in maintaining something in good working order
the work of providing treatment for or attending to someone or something
attention and management implying responsibility for safety
judiciousness in avoiding harm or danger
a cause for feeling concern
an anxious feeling

care (verb)	feel for (verb)
to hope, to desire or to prefer to have something, or to do something	share the suffering of
feel concern or interest
be concerned with
be in charge of, act on, or dispose of
provide care for