A friend and I were just talking about this last night, though in the context of USS Washington. Here is what we came up with:
Bismarck advantages:
Thicker armor and better subdivision
Faster
Better optics
Heavier secondary armament in (lightly) armored gunhouses
Slightly faster (but not really enough to matter) rate of fire for main armament
North Carolina Advantages:
Much heavier main armament (see below)
Very rapid fire secondary armament (remember, you do not have to penetrate armor to do damage topside – as Kirishima learned)
Better radar fire control
Better protection for fire control
Much more reliable machinery
Cruising range more than twice Bismarck's (17,500 nmi vs 8525 nmi), meaning North Carolina can steam at speed a lot longer than can Bismarck
Bismarck's 15" threw an 1800lb AP shell at about 2700 ft per second out to about 22 miles. That's a broadside weight of 14,400 lbs.
North Carolina's 16" threw a whopping 2700 lb shell at about 2300 ft per second out to about 23 miles. That is a broadside weight of 24,300 lbs. I give this to be a pretty serious advantage, considering Rodney worked Bismarck over pretty good with her 2050 lb 16" shells.
Bismarck could deliver (at maximum ROF) 48 100lb secondary shells, or 4800lbs of secondary armament per minute.
North Carolina could deliver (at maximum ROF) 150 55lb secondary shells, or 8250lbs of secondary armament per minute.
The issue then really is can Bismarck get a shell or two into North Carolina's vitals before those 2700 pounders level everything above Bismarck's armor belt.
My money would be on North Carolina.