When buying an AR-15 barrel, the only difference between 5.56mm and .223 Rem is in the CHAMBER specification. 5.56mm and .223 Rem bullets are both .223 caliber.
A .223 cartridge and a 5.56 cartridge are externally dimensionally the same. BALLISTICALLY they are different. 5.56 military loads are typically for higher muzzle velocities (MV), while a .223 Rem load conforms to SAAMI specifications.
The higher 5.56 MV requires a hotter powder load, and to avoid excessive chamber pressures, 5.56 chambers have a tad more expansion room at the shoulder and neck of the chamber, and there is also a bit more unrifled barrel area just ahead of the chamber, called the leade, in an effort to reduce the chance of pressure spike which can result if the bullet is already engaged into the rifling when the round is fired.
Unless you are expecting to use the rifle for match shooting, it is a better deal to get a 5.56 chamber, since it is OK to shoot both 5.56 ammo and commercial .223 ammo in it, whereas conventional wisdom says you should NOT shoot 5.56 ammo in a .223 chamber, for the above-mentioned reasons.
HTH