The extinguisher has a suitable role on a ship, I normally like to see both the extinguisher and the chemspray if a ship has two engineers.
Where the extinguisher really shines is when one is on fire, but not currently under fire. This is predominantly when facing a Hades/Incendiary ammo, or if your ship manages to get out of range of a flamer. The low repair cooldown allows for much faster fire extinguishing and repair than the chem spray. It is a reactive tool, as opposed to the proactive chemical spray.
Times that the Chemical spray shines:
-Hull (Pre combat), Guns, Pyra Balloons.
Times that the Extinguisher shines
-Hull (Post combat), Engine fires, Other Balloons, Mobula hull/balloons.
By far, using the two in conjunction is best. When under direct flamer fire (and unable to fire) the extinguisher should put out heavy fires and the chemcal sprayer should immediately be applied when the cooldown ends to grant the fire immunity buff. This, naturally, requires quite a bit of coordination, but when pulled off can completely shut down a flamer attack.