Old country (Jimmie Rogers, NLCR, Charlie Poole, Norman Blake, fiddle tunes, traditional bluegrass...)can be called "folk", but all folk is definitely not "country". Country is a sub-catagory of Folk.
I wouldn't call Jean Redpath, Planxty, Alan Mills, Mississippi John Hurt... "country", but they are certainly "folk".

On the other hand, I wouldn't call what passes for country these days "country" nor "folk".