Well, each caliber has its own merits and limitations. Let's review:
.223 - Also known as 5.56x45. This round can use bullet weights typically in the range of 40 grains to 69 grains. The 55 and 62 grain loadings are the most popular. The 55-grain loading is typically effective to 300-400m. Heavier loadings in the 62+ grain loadings can be effective out to 600m. For most practical applications, 400m is the maximum useful range.
This round at short ranges tends to fragment and break inside soft targets. This produces a terrifying wound channel and reduces overpenetration. Hollow and soft point bullets will typically expand and/or fragment to the extreme. Penetration can be iffy against hard targets and on frontal head shots. Against human targets and small game, this round is very effective.
.308 - Also known as 7.62 NATO. This round uses bulet weights typically in the 147 to 200-grain class. The 168 grain match loads are very popular, as are the 147 grain FMJ and 180-grain match millitary loads, respectively. Typicall effective range is 600m for most loads, although some rounds can be effective to over 1000m.
This is the premier intermediate-range precision round and is available in a staggering array of loadings from small game to ultra-heavy match loads. Penetration in humans is great, which can be a concern in some situations. Penetration of hard targets is very good, so the round can pound its way through light armour, trees, and moderate cover. This is the US Military's preferred sniper round (although there are numerous .50 caliber and .300 Winchester Mag rifles in service).
As to which is better, the answer is "Better at what?" For long-range precision shooting against game, persons, and targets under light cover, the .308 reigns supreme, but at a price. The round is heavier, recoils more, and may overpenetrate. The .223 is an excellent close-range round which is also effective on small game and persons out to 400m.
My $0.02 (this is the strictly opinion part):
For short-range defense and varmint work to 300m - .223. For ranges over 300m and for game larger than coyote, .308.
Hope this helps!