It's all about effectiveness at range, and viability on automatic fire.
.303 is effective beyond 1000m. Definitely a choice for a sniper weapon, but the ridiculous muzzle energy out of a .303 weapon makes it all but impractical for a personal weapon with automatic fire.
7.62mm (.308 as you called it) is effective to several hundred meters (I've heard up to but not beyond 1000). We use 7.62mm in general purpose machine guns, as they are either mounted or fired from a supported position. In personal weapons, note that the FN-FAL (aka, C1, C2), the C1 had no auto-fire, but the C2 did, as the C2 was designed to be fired in a supported position. 7.62mm is just not viable for a personal weapon with automatic fire.
5.56mm (.223) on the other hand, trades off muzzle energy, though the smaller projectile gives it a higher impulse. 5.56mm rounds are effective to a few hundred meters, making them practical for short to medium range engagements. 5.56mm weapons are more controllable under automatic fire, so in a pinch, a soldier can spray-and-pray a few rounds down range using an assault rifle with some accuracy, without having to worry about accidentally turning their air support into Swiss cheese.