Yes it's a different caliber, the 5mm in fallout is fictitious as far as I know.
5.56x45mm is also known as .223 though there are two main different types:
M193: was the original American .223 cartridge used by the M16A1 and Stoner rifles; most civilian rifles are chambered for this round.
SS-109 or (US) M855: developed by the Belgians as an improved version of the .223 cartridge; requires tighter rifling and the projectile is much heavier than the M193 thus achieving much better accuracy at longer ranges. For this reason it was officially adopted as the NATO standard and it is now widely used my most modern assault rifles.