They are the same cartridge, however the 5.56 rifles use a 1-8 twist (I think) and a 223 Rem uses whatever the manufacturer decides. The military are not bound by SAAMI, where the private sector manufacturers are. The two cartridge drawings are identical, however leade dimension can vary with manufacturers and with the military. The 5.56 has evolved from it's initial inception using a 55 gn bullet to I believe a 69 gn bullet with a sharper twist to stabilise the longer bullet. This evolution was due to it's ineffectiveness in Vietnam with the light bullets. Factory loads for the private sector manufacturers can run from 30 gns to whatever as long as they don't exceed the SAAMI spec'd pressures, again the military is not bound by SAAMI and set their pressure limits to function flawlessly with their semi and fully automatic weapons.
Neither one is better than the other, the military version is a specific duty loading and the 223 is a sporting version with many different loadings.