What's the distinction between "nonessential" and "inessential"?

Non-essential means not absolutely necessary, whereas inessential is more disapproving and can connote trivial.

According to Oxford Dictionaries Online, the prefix non- expresses a neutrally negative sense when added to adjectives, especially when the prefix in- has a special (I would say strongly negative) connotation. Another example is non-human (= not human) and inhuman (= cruel).

http://oxforddictionaries.com/definition/non-?q=non-

According to Merriam-Webster:

inessential, adj. 1 : having no essence, 2 : not essential; unessential

nonessential, adj. : not essential : not of prime or central importance

If the text is attempting to convey unimportance, then nonessential is the choice. If, however, the text is saying that something has no essence, then inessential is the word to use.

Edit: Note that the noun forms of inessential and nonessential have the same definition: something that is not essential.

The non- prefix expresses negation as an innate quality while the un- or in- may express negation as an acquire quality. It seems however this nuance is very weak and varies depending on the semantics.

For instance:

"His activities are unproductive, you should hire someone more qualified."
"His activities are non-productive, he merely is a consumer."

Perhaps more clearly:

"His work is non-scientific, he is a pizza-deliverer."
"His work is unscientific, he has no proof to support his evidence."

Another example:

"This file is unreadable, it was damaged"
"This file is non-readable, only a specific software can make sense of it."

The nuance with -essential however is very vague, in my opinion, because something is either essential or it is not. You could argue something is not essential enough and therefore "inessential" against something that absolutely isn't essential and therefore "non-essential", but I cannot think of any example where that would make sense.

It also seems that the use of those prefixes sometimes depend on the part of speech the word belongs to:

non-human (substantive)
inhuman (adjective)

It seems also seem to depend on the word derivation--but I am not so sure about that.