Fog is a form of a cloud and it is said clouds are visible moisture. We all know that mist (baby rain) is also moisture so I am curious when will a METAR say BR (mist) or FG (fog)?
The main difference between the fog and mist is their density and as a consequence, the visibility. From UK Met office:
... they are two distinct terms for a similar phenomenon. Visibility less than 1,000 metres we call 'fog' and obscurity with visibility greater than 1,000 metres we call 'mist'.
This is the same one given in FAA Aeronautical Information Manual (Page 7-1-62 in Dec 2015 Version). For visibility below 5/8 statute miles (~1000m), the term fog is used, while for values above that, mist is used.
Image from FAA AIM, page 7-1-62