is it true to say that 亡くなる is a polite form of しぬ?
Like what is the difference in nuance between these two sentences:
亡くなる is a euphemism for 死ぬ. 死ぬ means “to die” directly, and it is often appropriate to replace it with the less direct 亡くなる when one is talking about the death of a person. As Dave wrote in his answer, euphemisms for death exist in other languages, too.
死ぬ can be used for a person or an animal, but 亡くなる is usually used only for a person.
逝去する, mentioned in Derek’s comment on Dave’s answer, is a 尊敬語 (respectful form) of 死ぬ/亡くなる and raises the status of the person who is mentioned. 亡くなる is not a 尊敬語. For example, it is correct to use 亡くなる when talking about the death of a family member to the outside, but it is incorrect to use 逝去する in this case.
Same exact nuance exists in most languages, English included:
While you could say "I heard that your grandpa died", saying "I heard that your grandpa passed away" would be consider much better form.
Being the rudeness level, I think it's also a matter of shock level. Characters in anime and movies are likely to use '死ぬ' because it's very shocking to say it like that, where in real life people are a little more likely to have some restraint.
Let's face it, "お前はもう死んでいる" has more impact than "お前はもう亡くなっている". (Both mean "You are already dead", but the first is a direct quote from the anime Fist of the North Star's Ken.)