Well yes, nonessential elements are definitely always nonessential...but I feel like we're getting caught up in the semantics of these words and not really identifying the clear difference. For the sake of this answer, let's just use the phrase 'extra information' instead of 'nonessental elements.'
So, we have a sentence, with a subject (person or thing) and a verb (action). So someone is doing something. Example: The teacher yelled at the students. Now there are two types of 'extra information' that can be added here: extra information about the teacher (the subject or noun), or extra information about other stuff. So basically, an appositive always gives that 'extra information' about a noun, or the subject of the sentence.
Example of appositive: The teacher, WHO HATED HER JOB, yelled at the students. The core of the sentence is still just "The teacher yelled" but the extra information in between the commas describes that teacher. Appositive.
If, on the other hand, that extra information is not related to that understandably annoyed and jaded teacher, then it's just plain extra, or, yes, nonessential.
Example: The teacher, while the two boys held hands in the corner, yelled at the students.
See? Just extra information. Essential? No. Entertaining? Absolutely. But it has nothing to do with the teacher, and everything to do with the boys. Teacher: Appositive. Boys: Nonessential.
Hope that helps.