shooting vs firing off

shooting and firing off both are nouns.

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

As nouns, firing off is a hyponym of shooting; that is, firing off is a word with a more specific, narrower meaning than shooting:

Other hyponyms of shooting include shoot, countershot, discharge, firing, gunfire, gunshot, headshot, shellfire, potshot.

shooting (noun)	firing off (noun)
the act of firing a projectile	the act of discharging a gun
killing someone by gunfire