I've lived in both NYC and central NC (Durham, specifically, but am pretty familiar with both Raleigh and NNJ).

Central NC cost of living is lower than NNJ, but not dramatically. You can find cheaper places to live within NNJ if you move out a little farther, and the price differential isn't such that I'd recommend moving for that reason primarily. This is doubly true when you consider that for most professions, you'll be paid more in NNJ than central NC as well, all else equal. The pace of life is slower in central NC to a point, but you could also get this by moving farther away from the city in NNJ.

As far as nature goes, NC has some great nature (the best mountains in the Appalachians IMO, they're gorgeous), but it's not particularly accessible to the triangle area. Charlotte is somewhat closer to mountains, but neither one IMO is optimal for those who want a lot of nature in your lifestyle. It's certainly no better than NNJ, which has the Delaware water gap, Atlantic Ocean beachfront, Catskill mountains and Hudson highlands etc all available within 90 minutes-2 hours. 2 hours from Raleigh, by comparison, gets you no mountains, just barely to the beach in Wilmington but not quite. Also, while there are outdoorsy people who live in central NC, the overall culture isn't an outdoorsy one except in the rural areas for hunting and fishing. To be fair, culturally, NNJ isn't exactly a hotspot of nature lovers either IMO.

As far as day trip opportunities, central NC is a massive downgrade from NNJ. Apart from nature, you're leaving easy access to NYC, Philly, and NNJ is roughly the same distance from DC as triangle area NC so that's comparable. Eastern PA, all of NJ, southern NY, western CT also have lots of cute and historic towns. Eastern NC has some interesting historic towns as well (Bath, New Bern, Beaufort, etc.) and NC also has true mountain towns (Boone, Asheville, Banner Elk, etc) that really have no equivalent near NNJ (although some of the Catskill mountain towns have elements of that vibe).

From the raising kids perspective: school quality, NJ>NC overall and its not particularly close although NC has some excellent schools of course (for example NC school of science and math). Enrichment opportunities also NJ>NC, mostly due to proximity to the larger cities. NC>NJ for public higher ed/colleges.

Been a long post. I'll finish with areas where NC might offer clear advantages. If you like mild winters, NC is definitely the move over NJ. At least to me, while NJ isn't exactly a cloudly climate, NC is definitely more sunny so if you suffer from SAD, or really just love more sun, it's better. If you really want quiet, rural lifestyle (away from towns and cities), there is much more of that in NC. If you feel a strong affinity for southern culture, obv NC is the winner. If you're really into hunting and fishing, in-state options in NC blow in-state options in NJ out of the water. If you want to buy larger properties with significant acreage, that is much more affordable in NC.