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.

Worth noting that in the original 1970s comics written by Kirby that the New Gods are, literally, Gods, not aliens. Later (inferior) interpretations have made them aliens, but Kirby literally describes them as the new race of Gods that rises up after Ragnarok. i understand the "New Gods are aliens" canon is the current modern canon but it's not the canon that Kirby wrote and stating otherwise is inaccurate.

However, the Old Gods are not our classic gods.

It's heavily implied it's intended to be the Norse gods- we see the old ruins a few times, notably in Lonarr's short, and in the flashbacks you can see Thor. This was Kirby building his own universe and he had always intended the New Gods to be the successors to Thor's adventures.

Mark Evanier's 'New Gods' also emphasizes the Old Gods being the Norse Gods. It's not Jack Kirby, I know, but I think he expanded the mythos pretty well.

I was more talking about the Ancient and Roman gods, as that pops up. A lot.