Hurricane Damage

My home received damage during Hurricane Sally and the field adjuster wants me to replace the old school acoustic ceiling tiles in two rooms with new acoustic tiles. Problem is, I can't find any contractor in this area that install acoustic tiles.

My question for you guys is, do they actually offer any acoustic benefit over drywall?
 
Here is a photo of the current acoustic tiles.
 

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Sure.
Acoustic ceilings (Rockfon e.g.), certainly when applied with enough rear volume absorb broadband across the full spectrum. Drywall acts as a panel resonator and will only absorb certain frequencies defined by the resonance of the system.
Now I do not know where you live, but acoustic ceilings are quite common, in fact a lot of contractors don't even realize they are acoustic ceilings.
 
Acoustic tile is much, much better then drywall/gypsum. In the USA the most common brand is Armstrong and there is a wide choice at both Lowes and Home Depot. It's the tiles commonly used in drop ceilings.
 
I don't know if the ones you have are still made. I had a room in my old house with the same as your photo and they were glued directly to the ceiling. It's worth a look at Armstrong to see if the stick on type are still available.
 
One thing to add, I am in Pensacola Florida so directly above our ceilings is the attic and is covered in blown in insulation.

If I made the ceiling all out of mineral fiber, is all my air conditioning going right outside. Our attics have gable vents to the outside air.