The difference in use between utan and men, which also translates but, is a matter of whether the content of the "but" clause is considered as something contrary to the content of the preceding clause or considered as something that partly has a similar meaning or function in the context for those involved. "That dog is not black but dark brown" would be translated "Den hunden är inte svart utan mörkbrun" (http://en.wiktionary.org/wiki/utan)
"Utan" is used as a more of an opposite, whereas "men" is used as an any. A good example is, "Hunden är inte svart utan vit" because "Svart" and "Vit" are opposites, like walking(att gå) and running(att springa). With "men", a good example is "hunden är inte svart, men djupblå"
I think there is a difference. I would think of rather as a more preferred state and would translate it to "hellre" or "snarare".
"Jag springer hellre än går." - "I rather run than walk."
"Det är ingen igelkott, snarare en sten." - "That's no hedgehog, rather a stone."
"Det är ingen igelkott utan en sten." - "That's no hedgehog, but a stone."
The last two sentences are quite similar, but there is a difference in certainty. In the last case the speaker is certain that it is a stone, whereas in the other he think it looks more like a stone than a hedgehog, but it could still be something completely different.
Rather has multiple meanings and is not always a synonym for prefer.
You could certainly say "That's no hedgehog. Rather, it is a stone." This would work even if you would prefer it was a hedgehog.
I think the answer to the previous question is yes, utan can sometimes translate to rather. But I'm not fluent in Swedish so I'm willing to be corrected.
Arnauti explained it well in another sentence. Utan can be 1. a preposition meaning without - Jag dricker te utan socker. I drink tea without sugar. 2. a conjunction which requires the condition "Not X, but Y", so there must be a negation before "utan". Jag älskar inte dig, utan honom. I do not love you, but him.
It depends on the context. In the sentence above "Han går inte" has to mean "he is not walking" or "he doesn't walk". This is because the ending "utan [han] springer" tells that he is really running which wouldn't make sense to replace most translations of goes with.
Just "Han går." however would translate to "He is leaving" or "He is walking".
On the other hand "He goes..." would often be translated to "Han åker..." as in "Han åker buss till Stockholm" - "He goes by bus to Stockholm". But also here it depends on the context.
Here are I use "Han går" in a few different contexts
"Han går på fest." - "He attends a party." or "He goes to a party"
"Han går till festen." -"He is walking to the party."
"Han går på fester." - "He attends parties."
"Han åker till festen." - "He goes to the party"
"Han går an" - "He is acceptable"
"Han går in" - "He enters", "He goes inside" or "He fits" (size-wise)
"Han går ut" - "He is going out", "He goes out"
"Han går upp" - "He gets up", He goes up"
"Han går under." - "He goes to his doom."
"Han går under bron" - "He walks under the bridge."
Arnauti explained it well in another sentence. Utan can be a conjunction which requires the condition "Not X, but Y", so there must be a negation before "utan". Jag älskar inte dig, utan honom. I do not love you, but him. Or as in this sentence: Han går inte utan springer. He is not walking, but running.
I tried "He doesn't walk, he runs" which I think is a more satisfying translation in English. The 'but' is superfluous. I was marked down. Not sure if there's a solution to this as the men/utan distinction is important to teach but it might be hard to think of sentences that force an English speaker to recognise the difference.