What the sniper is shooting is also generically speaking a ".22 caliber", but it is a centerfire cartridge with a separate primer installed in the bottom of the case. The case expands outwards from the neck in the shape of a bottle to hold much more powder. The M-16 and AR-15s etc typically fire a 50 to 65 grain slug at from 2900 to 3200 fps depending on the load. At these extreme velocities the wounding and killing power is MANY times greater than the 1000 fps .22LR.
There are other ".22 calibers" even larger in case size than the .223 the DC sniper is using, which push rounds up near to 4,000 fps. These are generally used for varmint hunting and target competition.