The 5.56 NATO chamber is a bit looser in the body than the .223 Remington (SAAMI) chamber for reliability and has a much longer leade (throat) than .223 Remington.
.223 Wylde was meant to give a slightly tighter body spec than 5.56, but a bit looser than .223 Remington, while retaining a longer leade/throat to optimize the use of longer bullets, specifically 80gr, which are prevalent in High Power Service Rifle competitions.
There is a slight headspace difference, and while SAAMI recommends against 5.56 NATO ammo from being fired in a .223 Remington chamber, I personally don't think this is an issue with AR-15 rifles. But stick to SAAMI recommendations for your safety.
As far as 5.56 vs .223 Wylde, if you are going to shoot High Power, get the Wylde. Otherwise, just stick to the 5.56 NATO. You aren't going to gain much if all you plan on doing is using factory commercial, USGI spec ammo, or just plain handloads with 77gr or lighter (non-VLD) bullets.
5.56 NATO actually has a longer leade/throat than a .223 Wylde.
If you want to shoot 80gr+ BTHP or 75gr+ VLD-style, then get the Wylde. You can shoot 80gr (or heavier) BT or 75gr (or heavier) VLD in a 5.56 NATO, but the leade is going to be longer than you want it to be, and to seat to the lands will probably mean you're already pretty high in the neck to begin with. After 1500 rounds or so, the throat will wear out quick enough that you might no be able to seat to the lands anymore. Thus, the Wylde will be the choice.