I am an aviation meteorologist and I struggle to distinguish the two of them.
The term haze does not say anything about the humidity of the air because it is by definition caused by dry particles, so not by water. If you have mist (or fog), then water droplets are suspended in the air, which usually comes with 100% humidity.
Here is a set of definitions by the UK's Met Office:
In our meteorological glossary fog is defined as 'obscurity in the surface layers of the atmosphere, which is caused by a suspension of water droplets'.
By international agreement (particularly for aviation purposes) fog is the name given to resulting visibility less than 1 km. However, in forecasts for the public, this generally refers to visibility less than 180 m.
Mist is defined as 'when there is such obscurity and the associated visibility is equal to or exceeds 1000 m.' Like fog, mist is still the result of the suspension of water droplets, but simply at a lower density.
Mist typically is quicker to dissipate and can rapidly disappear with even slight winds, it's also what you see when you can see your breath on a cold day.
A third term you might also hear mentioned is haze. This is a slightly different phenomenon which is a suspension of extremely small, dry particles in the air, not water droplets. These particles are invisible to the naked eye, but sufficient to give the air an opalescent appearance.