shooting range vs firing range

shooting range and firing range both are nouns.

Word	Noun	Adjective	Verb	Adverb
shooting range	Yes	No	No	No
firing range	Yes	No	No	No

As nouns, firing range is a hypernym of shooting range; that is, firing range is a word with a broader meaning than shooting range:

shooting range: an enclosed firing range with targets for rifle or handgun practice
firing range: a practice range for target practice

Other hypernyms of shooting range include target range.

shooting range (noun)	firing range (noun)
an enclosed firing range with targets for rifle or handgun practice	a practice range for target practice