Hola Emma

In weather expressions, "hace" is generally used in phrases that talk about temperature or suggest temperature. Here is a specific lesson on hace. 

Hay is also used in weather phrases but not necessarily expressing temperature. It's mainly used for fog, thunder, storms... 

Bear in mind these are idiomatic expressions that don't necessarily have the same grammar as in English, so they need to be learned and memorised. 

Saludos