At first : The other answers already showed the existance of such a deadly weapon, mentioning the Girandoni.
I would like to write about the disadvantages.
I think that you should take armor into account.
Finding armor that is strong enough to protect you from being killed by a bullet fired from an airgun is far easier than finding armor to protect yourself from a firearm.
Also, in fights involving masses of people, a firearm is very likely more useful;modern assault rifles could fire bullets piercing through up to 3(?) people - your airgun would very likely only kill at most(as it's not very lethal from what I found, see below) one per shot.
Furthermore, I consider regular bullets to be far more devastating; what if you would hit your enemy at - let's say - the knee ? The hit by the airgun wouldn't pose a big problem; but when shooting somebody's knee with the power of firearm, he would very likely not be able to walk anymore.
From Wikipedia "Projectiles" :
Projectile energy air gun : 11,000 to 29,800(air rifle, for example Girandoni)
Projectile energy firearm : 58,000(pistol) to 470,000(AR)
all values in J/kg
This means, a lightweight easy to use pistol has about twice the energy of a 4kg Girandoni. And an AR even got about 16 times(!) the firepower.
And from this article :
2,300 newtons of force would be needed to crush a human skull
So let's suppose the stop distance of the projectile is 6,5 mm(average male skull thickness).
We calculate F = KE ÷ 0.65 centimeters, where KE is one of the values mentioned above.
So for our Girandoni : 45846,15 J/cm are 458,46 Newton
And for the ARs : 458,46*16=7335,36 Newton
So as a rough estimation : The AR could crush more than one skull, while our air rifle cannot
Herein, take the skull only as an example. Think of somebody protection himself with thin steel plates. An AR could pierce, while your Girandoni would struggle.
Concerning weight, the Girandoni which has been previously mentioned is quite similar like a typical AR(M-16 is even less heavy). In this point, pistols are better.
The only advantage I see for air rifles is stealth.
As mentioned in other answers, they are much harder to spot than conventional firearms.
And if - in your post-apocalyptic world - still some remainders of society exist, like forensics or police, they may draw many conclusions from an advanced ballistic research - such a thing isn't really possible when you use an air rifle.
So in the conclusion, a firearm is reasonable if you have only a few poorly armored enemies and are good at aiming.