there two ways to achieve the snow motion blur.... if you want only the snow to have motion blur, just right cick the particles, select properties, the on the object properties that will appear go to general, down below there is motion blur, tick the item object. this will give a motion blur effect for the snow.
the other one is with camera.
select your camera, go to modify, down below enable multi-pass effect, then on a small window under multi-pass effect, select motion blur. this effect will give the snow a motion trail.