Page "Snap (gridiron football)" Paragraph 10

The game design rationale for requiring the snap to be a quick and continuous motion to the backfield is to eliminate the need for rules provisions for a live ball in scrimmage.

In Rugby Union the ball may be retained by the forwards and played for a time via the foot in a scrummage ( which Rugby League has as well ) or ruck, or by the hands in a maul, necessitating additional restrictions on play and player positioning during those intervals.

In American and Canadian football, the ball as it is put in play is only held in the line ( by the snapper ) for a fraction of a second.

The uncontested possession also, as Walter Camp pointed out, allows for better offensive and defensive planning by the side entitled to snap the ball and their opposition, respectively.

A muffed snap can be recovered by either team.