Why is ice more reflective than liquid water?
In fact ice is slightly less reflective than water. The reflectivity is related to the refractive index (in a rather complicated way) and the refractive index of ice is 1.31 while the refractive index of water is 1.33. The slightly lower refractive index of ice will cause a slightly lower reflectivity. In both cases the reflectivity is about 0.05 i.e. at an air/water or air/ice surface about 5% of the light is reflected.
Water generally has a relatively smooth surface so the light falling on the water only gets a chance to reflect back once. Any light that doesn't reflect off the surface propagates down into the water where it is eventually absorbed and converted to heat. The end result is that a large body water reflects only about 5% of the light.
Ice is generally covered with some snow, and snow is made up of small ice crystals with air gaps between them. Light falling onto snow may be reflected at the first surface, but any light that isn't reflected will meet lots more ice/air interfaces as it travels through the snow, and at every surface more light will be reflected. The net result is that much more of the light is reflected from snow.
So the difference isn't anything fundamental, it's just because water is continuous while snow isn't. It is possible to form an air water dispersion, for example foam or fog. Both foams and fogs reflect light far more efficiently than a large body of water.