Nevertheless and nonetheless mean the same thing (‘notwithstanding’), so they can be used interchangeably. Nonetheless is a more recent coinage.
Both started as three words, but over time ended up as one. It’s now always nevertheless, but the three-word form none the less is still seen, especially in the UK.
There is a certain logic to none the less, if spelling it as a single word might suggest an odd pronunciation (non-EATHE-less, with the second syllable rhyming with breathe).
But you’re fine nevertheless with nonetheless.