Why might ice and dry ice behave this way?
The ice melts into water, and the volume of water produced is the same as the original volume of ice.
The volume of dry ice decreases. Ice crystals form on the dry ice. After some time, a pool of water is the only remaining substance.
When ice melts into water, the volume of liquid water should actually be a bit smaller than the volume of the original ice; any given volume of water will create a volume of ice about 9% larger. This happens because the individual water molecules each get a little wider (the hydrogen atoms spread apart) when water changes to the solid form.
So-called "dry ice" is actually the solid form of carbon dioxide. When you let a block of dry ice sit at room temperature, the carbon dioxide sublimates, going from the solid state directly to the liquid state and evaporating away into the air. If you could see the gas phase of the carbon dioxide, you would realize that its volume actually increases greatly as it sublimates.
The reason you see ice crystals forming on the dry ice is because of condensation. The dry ice has to absorb energy from its surroundings in order to sublimate; this means that the dry ice and the area around get colder and colder. If there is any water vapor present in the air nearby, it gives up its heat energy and becomes a solid, resulting in the appearance of ice crystals. Once the dry ice has all sublimated, the ice crystals get warmer by gaining energy from the surroundings, and they melt.