American soldiers used netting over their M1 helmets in order to reduce their shine when wet or to insert leaves or pieces of cloth to act as camouflage. As /u/sandbagger69 noted, soldiers could often use the net in order to stow miscellaneous things, like cigarettes or in the case of paratroopers, the parachute first aid kit. The kit also could just be strapped around the helmet. The first pattern had a zipper. The second pattern, more common around the time of D-Day, was in a sealed canvas pouch.
Interestingly, the various kinds of helmet nets used by the Army were for the most part not standardized, but rather taken from British or Canadian stock, or cut from other miscellaneous netting. These nets varied in size from 1/2" to 3/4" squares.
The Army only got a standardized type of net with the introduction of the late-war M1943 uniform, that included boots, trousers, a jacket, and other miscellaneous improved items. This net had a much tighter weave (1/4" squares), and had an elastic band to secure it around the helmet. This net became common, along with various pieces of the M1943 uniform (the M1943 seemingly flowed in piecemeal; jackets show up first, then boots, and finally the pants) through the late summer and into the fall of 1944.
An outlier was the 79th Infantry Division, who used a much larger net (2" squares) resembling something normally associated with the Japanese. This type of net is only associated with this unit.
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