The .308 is used extensively as a designated marksman round. It is well-suited to this role. A skilled shooter can accurately place the round into a target at very decent ranges, and do so with enough power to do considerable damage.
However, it was long ago determined that such power was wasted on the average soldier, who is not trained to such a degree of advanced marksmanship as is required to fully utilize the benefits of a "serious" round like the .308. Don't get me wrong, I'm sure most of us here can use it, and it seems easy. But remember that a great many fresh recruits know nothing about guns, or marksmanship, or even basic shooting.
The most important criteria is getting the most soldiers into the field, as effectively as possible, as quickly as possible, for the least amount of money possible. The heavy battle-rifle rounds never succeeded in this role. On the other hand, the .223 is almost trivial to put into the hands of an inexperienced shooter and have him using it effectively in a matter of days (if not hours!). The .223 is nothing to sneeze at, and it has enabled our armed forces to function as only a forced composed entirely of elite soldiers could have functioned in the days of the battle rifle (today, our foes must hide their heads constantly, knowing that essentially every single soldier they are fighting is capable of taking a head shot at surprising ranges – a fact that are not capable of replicating within their own forces).
The .223 round is easily controllable in burst fire, a combination which, it was found through empirical studies, leads to the highest hit ratio per round fired – higher than any rate of fire for heavier rounds like the .308. Not only that, but its flat-shooting nature and mild report make it easy for beginners to make shots at good ranges, without being distracted by all the fuss the gun itself is making.
There are very good reasons why the .223 has replaced the .308 for the majority of our troops, and why the overwhelming majority of armed forces around the world have made similar changes (see the AK-74 for instance). I suggest you buy a book on the subject; I'm not the best source of such info.