What determines the weather in Minecraft?

Solution 1:

Yes. In an area covered by snow, it will only snow. Weather is controlled by the type of biome it is affecting. It doesn't rain in Tundra, Taiga and Desert biomes.

At the moment, there are two types of weather: Rain and snow. Thunderstorms can occur in both types, and these can be dangerous: striking things with lightning (including you!), creating fires (as a result of the lighting strikes) and etctera. The occurrence of weather lasts for 15 minutes, if you don't want to go through it, you can sleep in a bed to skip that time entirely.

As a proof that different biomes have different types of weather at the same time, see the following image: (courtesy of the Minecraft Wiki)

More information about weather: http://www.minecraftwiki.net/wiki/Weather