5.56 is loaded per mil specs. .223 is loaded per SAAMI specs. Mil spec is about 50k psi where SAAMI is about 45k psi (dont quote me on this, but it should be close.....) Anyway, the 5.56 is a little bit hotter and is also about .002" smaller in one of the diminsions to make Mil Spec for "field" gage (it fits into dirty chambers).
Shooting a 5.56 round in a .223 will give you a little headspace issue and a little pressure issue. Most noticable would be in a bolt action rifle where there is not pressure being used to work the action. You will feel the difference in recoil between the two, however slight it is.
Brass for .223 made per SAAMI spec will say ".223" on the headstamp. If you have a case that does not have ".223" in the headstamp, it is likely a military or Mil Spec 5.56 brass. Brass for mil spec can be thicker, not always so, but can be. It will also likely have crimped in primers so a loose primer will not cause a jamb in a semi auto.
Many people use .223 and 5.56 brass interchangably when reloading. With most loads, most people cannot tell the difference. Getting into the max end of book loads is DANGEROUS if mixing brass because a load with one brand of brass may not have the same pressure as another brand. You want to work up a load using the same headstamp brass if possible. If you are shooting a mild load, mixing is usually ok.
Actual 5.56 mil spec brass is usually not reloaded for the military. Very few people have the correct dies. Small Base .223 dies will work for 5.56 NATO because they can size the brass down far enough to make spec so some military folks or military marksmen will load up to 5.56 NATO Mil spec for their specific firearm. If you do not have a military firearm in 5.56 NATO, do NOT load per 5.56 book loads.
The NEF rifle is plenty strong for either .223 or 5.56 ammo. 5.56 brass is going to stretch in your .223 chamber on it's first firing. Full length sizing several times will cause this stretching to occur several times. After about 3-4 reloads, you could have a case head separation which is not a good thing.
.223 brass is a few thousandth longer and will not stretch as much. Again, full length sizing for your NEF will cause excessive stretching and can eventually lead to case head separations. Probably not as fast as the 5.56 brass because .223 brass is longer to start with and is a little softer.
Best way to load for your NEF is to get 100 brass with same headstamp. Does not matter which headstamp, just get .223 or 5.56 once fired brass of same headstamp. Set up your sizing die so this fired brass just fits into your NEF chamber. You want to have just a touch of tension as you close the action. If you are sizing your brass to fit your action you will not have as much brass growth and your brass will last longer and you will be less likely to have any head separation. Work up a .223 book load with this brass and you are good to go.
Lots of folks will get neck sizing dies so brass is not excessively sized, fits chamber better after first firing, and gives better consistency thus probably better accuracy. The average shooter can accomplish primarily the same thing by setting their full length die to size the brass and fit their chamber.
I have some Federal American Eagle and some white box Winchester 5.56 Mil spec ammo that I have fired in my NEF and in my Contender. It is not as accurate as .223 ammo, but it shoots just fine. My fired brass is put into boxes so this brass stays with the specific gun it was fired in. I will adjust my sizing die so this brass will stay sized for the specific chamber it was fired in. I have some brass that has been fired 10-12 times with no problems. Case neck splits almost always occur before case head separations if cases are sized in this manner. Once you get 5-10 cases with neck splits or if you see a ring starting to show on the walls of your brass, it is time to discard all 100 brass and start over.
Good luck with your loading. Stay within book loads, and size your brass for your gun.
Steve