The difference between mist and fog is visibility (how far you can see through it). If visibility is less than 1,000 metres we call it 'fog' and if visibility is greater than 1,000 metres we call it 'mist'.
We call this visibility and its an important part of forecasts affecting many aspects of life from driving conditions to shipping and aviation.
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 the term in forecasts for the public generally relates 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) which are invisible to the naked eye, but sufficient to give the air an opalescent appearance.
These particles can also contribute to creating a red sky at sunrise or sunset.