Why does weather exist? How did rain and snow exist or begin?
It is the movement of the atmospheric gases around the planet that creates weather. Places like the Moon or Mercury do not have weather - only temperature changes. Other planets like Venus, Jupiter, Saturn, Neptune, and even Mars to some extent have "weather patterns" of different winds, precipitation and temperatures.
The sun's energy powers the planets. The distribution of that energy around the planet is effected to a great extent by the atmosphere. The energy transfer is what creates winds, relative differences in pressure, and drives the atmospheric part of the hydrologic cycle.
Rain and snow are both parts of the hydrological cycle.