special interest vs special-interest group

special interest and special-interest group both are nouns.

Word	Noun	Adjective	Verb	Adverb
special interest	Yes	No	No	No
special-interest group	Yes	No	No	No

As nouns, special interest and special-interest group are synonyms defined as:

special interest and special-interest group: an interest group that tries to influence legislators or bureaucrats to act in their favor, typically through lobbying

Other synonyms of special interest include advocacy group, lobby, lobby group, pressure group, special interest group, third house.

special interest (noun)	special-interest group (noun)
an interest group that tries to influence legislators or bureaucrats to act in their favor, typically through lobbying	an interest group that tries to influence legislators or bureaucrats to act in their favor, typically through lobbying