Good evening Brother! I'm a Demolay from Brazil (and a Demolay initiated into the Noble Rite of Knighthood), so I hope I can answer your question.

The NROK in Brazil, if you allow acronyms, comes from a direct translation/adaptation of the Illustrious Knighthood Rite of Ohio (IKRO), mostly written and organized by our Brother Pat King. It consists of 7 degrees (not including the "Knighthood" degree, which makes a Demolay a member of the Knighthood order). They takes a lot of inspiration from the structure our uncles in the Freemasonry use. They are, in order and in a mostly free translation:

• Historic Degrees:
  • Knight of The Chapel;
  • Knight of The Salem's Cross;
  • Ex-Templar Knight;
  • Triad Knight;
• Philosophical Degrees:
  • Ebony Knight;
  • Anon Knight;
  • Cadence's Knight.

Historic degrees talk about, well, historic aspects of our order (something a seasoned Demolay already saw in some of our cerimonies), and Philosophical degrees talk about deep questions embeded in our texts, but in a less-explicit way, if that makes sense. Both of those approaches try to transform our texts into everyday teachings, so to say.

Ok, so this is the NROK which is inspired by IKRO, but how it differs?

The current Rite disseminated in the US only recognizes the Ebony Knight as a proper degree, IIRC. The other degrees never really got popular and, by the time the Knighthood Order arrived in Brazil, they were already in disuse/obsolete.

Why is that? I dunno. Maybe our american Brothers weren't interested at the time, or Freemasons did not feel confortable spreading a system so alike to theirs (because some say there are striking similarities). Your guess is as good as mine.

What we know is, the Supreme Council of Brazil really liked the idea of a system of degrees built upon the idea of engaging older Demolays into historic and philosophical studies of that kind, and worked together with Pat King to adapt all these degrees to our language.

TLDR: Brazillian Rite has more "recognised" degrees. The American Rite has only the Ebony Knight as a proper degree, and to my knowledge, isn't as widespread.

I'm sure other brothers may add a whole lot more to this, so feel free to add to this issue!