Having played field hockey for much of my life, I never really thought about why field hockey sticks are so stiff compared to ice hockey sticks and what implication this has on shooting in both sports. The fundamentals of physics suggest that the ball velocity of a field hockey shot is created slightly differently than the puck velocity of an ice hockey shot.
Since field hockey sticks are so hard and stiff, the stick is minimally flexed before it makes contact with the ball. When hit, a field hockey ball gains its velocity predominantly by the kinetic energy created as the player accelerates the stick. A large force is exerted on the ball and momentum is transferred from the stick to the ball.
However, in ice hockey, the puck velocity on a shot is created through kinetic energy and elastic potential energy. While the puck gains velocity through kinetic energy in a similar mechanism as a field hockey ball, a puck gains additional velocity through the release of potential energy that is stored when the hockey stick is flexed against the ice. The momentum created by the velocity of the moving hockey stick and the release of the stick flex is transferred from the stick to the puck and determines the velocity of an ice hockey shot.
Another reason why ice hockey is a much faster paced sport is because of the friction involved. Field hockey is commonly played on grass turf, which, compared to ice, is much rougher. Assuming hockey pucks and field hockey balls are made of the same type of plastic, the coefficient of friction for the between the turf and ball is much greater than that between the ice and puck. Thus, if one was the apply the same amount of force to both the puck and ball, the puck would travel much farther. This is why the field hockey turf is watered at the beginning of the game and a halftime, as water helps to reduce the friction between the ball and turf.