Both mean the same thing in Norwegian. You can say “gå tur i skogen” or “gå på tur i skogen” (go for a walk in the forrest). However, you may use the second form in many other situations, such as “gå på kino”, “gå på biblioteket”, “gå på do/toalettet”, “gå på kafé” (places), but also “gå på besøk”, “gå på universitetet” (attend university).
I believe that in Danish, there is a clearer distinction between 1 and 2. In Danish I believe the first form is preferred when you are speaking about going for a walk, but the second refers to “taking turns” (“de lot flasken gå på tur”/“they sent the bottle around”).