The difference is a subtle one, and the two can frequently (typically?) be used interchangeably. But what difference there is between these two compound words is rooted in the single components that make them different: never, and none.
"Never" is concerned with time, and so is more directly applicable in connection with an event or something that can happen: "It's cold today, but I am going out nevertheless." There's been no change in the likelihood of a given thing happening.
"None" is concerned with quantity, and so is more directly applicable in connection with a measurable quality, attribute, condition, degree, etc: "It's not as cold today, but it is cold nonetheless." There's been a change, but the change is being discounted, and so the given situation or condition remains the same.
Even in those examples, the two can be reasonably swapped. But, in imagining a context where those two sentences might be used, there's a shift in tone or emphasis related to an intention to go out; or an assessment of how much intention there is to do something (go out) that is not given in the sentence itself, but that in context is or will be understood -- nonetheless, or nevertheless, or despite that, or without consideration of that.