Something that I've known for a long time is that the area of the Peak District National Park and that of the North York Moors National Park is almost the same, 555 square miles, compared to 554.
This is certainly something the two areas have in common. I can't really think of anything else though; certainly the landscape, geography and culture is quite different - I suppose they both have, to a degree, a shared industrial heritage though.
The major differences: The North York Moors includes a section of coastline; more people live in, and visit, the Peak District; public transport access is much better in the Peak District; there is well-dressing and Hallamshire carolling in the Peak District...they might have similar cultural events in the North York Moors, but I don't know.
I've walked in both national parks...but obviously know the Peak District much better.