"Haze" is anything in the air that partially obscures vision. It's usually a light fog, but you can also have a "heat haze" when the air is hot enough to make things appear wavy. The word focuses on the fact that you can't see very well.
"Mist", on the other hand, focuses on the fact that there's some stuff in the air - usually water, in the case of regular fog, but it can be anything that's visible. If you use dry ice, it produces some carbon dioxide mist/fog.
"Fog" is the most common when talking about the weather, but "mist" is more common in many other contexts.