My husband (he's a native speaker but not a grammar teacher :) ) explained it as "ute" is a general term for outside whereas "utenfor" is a specific location outside. So if I asked, "Hvor er barna?" then he could say "Ute" or be more specific with "Utenfor låve."
He said you would never just say "utenfor." Hope this explanation helps!
Would it be correct to think of "ute" as "outside", and "utenfor" as "outside of". While it may not sound entirely natural in English, it would help to keep the difference clear in my mind, at least at my current level of understanding of those words. Much in the same way as "innenfor" meaning "inside of", or "within".