What's the difference between "shrouded with" and "covered with"? Any different hues of meaning here?
The difference is largely one of connotation. The verb to shroud derives from the noun shroud, which typically refers to a sheet used to cover the dead for burial in some religious traditions. Because of this association, when you say that something is shrouded with or shrouded in, it connotes an atmosphere of mystery, gloom, or the numinous.
Shrouded in also lends itself more readily to metaphorical usages. You might say that "The castle was shrouded in mist", but it would sound odd to say that "The castle was covered in mist".