From an English perspective--including mine since I'm learning Japanese using English, I felt that 違う sounded strong, this may be from the tendency to respect others' opinions on English, that implying others to be wrong may be seen as rude. But from the situation that I've seen it was used, and based on a native I asked; 違う was used freely like a 'no' in English when you disagree with the information being said. For example, if we were asked ”大学生ですか” the native Japanese that I asked said it's fine to reply with ”違います” if we aren't, and that it can be used in other formal situations. But another non-native yet advanced Japanese speaker claimed the word to be strong and can be rude. I know since I already got confirmation from a native speaker, it should be done, but I'm seeking more info, when is it fine to use 違う and can it be rude?
I rather read your remark on English connotation interestingly, but anyway, there do exist some cases where the superficial form and the meaning 違う, in any language.
Effectively, you can just take 違う as Japanese way to say "it's not", because:
あなたはプログラミングしますか？ [ordinary verb]
→ いいえ、プログラマーじゃありません。 (too long!)
→ いいえ、*じゃありません。 (ungrammatical!)
So this is our pragmatic solution for this specific linguistic problem, and not much related to anything cultural. Incidentally, in casual speech we sometimes simply say じゃない（です） in this context, which is obviously not a good grammar, but as short as 違います／違う in mora count.