Frost and snow are made of ice but form as intricate crystals whereas ice that forms on a lake or a puddle is more of a hard blob of water. The substances are the same. Impurities aside, water is water.
The difference between crystalline ice and plain ice is in the formation. Snow and frost are formed directly from water vapor in the air. As more and more water vapor adheres to the developing structure, the crystal begins to grow.
This is different from sleet or ice pellets, which are formed when liquid raindrops freeze in the air; or from ice on a lake or puddle, which is just frozen water. These kinds of ice do have a crystalline structure. Break a piece of ice and see how sharp the edges are. But the overall structure is less rigidly organized.
Sometimes a frozen puddle of water will have areas of crystals built into the overall plain ice. These are areas where water vapor in the air forms a crystal on the water ice.