Interesting concept, though I don't know if it would be much quieter than a conventional rifle.
Correct me if I'm missing an important detail, but rapidly expanding gases from any source will produce a an audible shockwave. In order to propel a projectile at speeds similar to the same speeds as conventional rounds, the compressed air would have to be expanding at the same rate and exerting the same pressure as a gunpowder explosion.
You're essentially correct. An explosion produces a sound because the rapidly expanding hot gas, resulting from the ignition of the gunpowder, accelerates the air around it at the speed of sound or more, so a gun which uses compressed air to fire a bullet would also produce a loud noise. I don't think it would be as loud as gunpowder, but it would definitely be a big bang. A better idea would be to integrate a silencer into the barrel of the gun.
Of course, snipers really don't need a silenced gun anyway, as they are usually trained in how to sneak in, pop the target, then sneak out without being detected.
Good observations all the way around. I had some powerful airgun in Britain that were quite loud. However, the sound can be mitigated substantially in the same way as a firearm suppressor. Perhaps this weapon integrates a way to slow those gasses after the projectile exits?
Hmm. That could actually keep the gun from firing correctly, as the gases would need to escape for the system to work, and the velocity of gases escaping at the speed needed to propel a bullet at lethal velocity would be such that any escaping gas would be heard.