A lot of SWAT and police units use .223 chambered rifles for precision work for 2 reasons.
1. Shots are almost always (if not always) less than 100 yards. The .223 is still effective at this range.
2. Penetration is reduced by using a smaller round, thus reducing the risk to innocent bystanders.
The military won't use .223 as a sniper weapon because it loses too much of it's terminal performance, especially once it gets past about 300 yards (it might make 400 out of a 26" barrel).
As far as what our guys are using, as pointed out, the Army uses the M-24 (a souped up Remington 700) while the Marines use the M-40 (originally based on either the Remington 700 or Winchester Model 70, I can't recall which). They also BOTH use the Barrett M82A1. The Marine Corps just call it the M82A1 while the Army is calling it the XM-107 (per the article in American Rifleman a couple of months ago).