What are the similarities between them?

Right. New Gods are effectively an alien race occupying two planets that exist in a dimension all their own. Lords of Chaos and Order are beings that are empowered by magic and embody the abstract nature of those ideas (chaos and order). People can become Lords of Chaos or Order. People usually can’t just become New Gods (except in the couple of cases where humans’ physiological structures were altered to artificially make them New Gods, but I digress).

Thank you for these answers, I was mostly focused on the more conceptual part.

One was a concept developed by Jack Kirby for his alien mythology, the other was developed by... I dunno, plenty of writers... as an overarching element to all (or most of) the magic on Earth. They don't have any overlap in form or function that I'm aware of.