General:
* Climate: This refers to the long-term average weather patterns in a region.
* Conditions: A broad term that can describe the state of the weather, but also other things.
* Atmosphere: Refers to the air surrounding the earth, which is where weather occurs.
Specific aspects of weather:
* Precipitation: For rain, snow, sleet, etc.
* Temperature: For hot/cold conditions.
* Wind: For the movement of air.
* Sky: For the appearance of clouds.
The best synonym depends on what you're trying to communicate.