Fog is simply brouillard, something that is hazy or difficult to see through because of a cloud. Murk implies more of a muddiness, where the haziness is more of a darkness or a dirtying.

Fog is more of a physical thing, low cloud, essentially. Murk is more of a perceptual thing, meaning gloominess or darkness, and there may also be an element of haze or fog. The net effect is making it hard to see and make things out. Of course, if a fog is thick, it can become murky. Both terms can be used figuratively, too, in which case they have pretty much the same meaning of lacking clarity.