Late to this thread, but thought I'd chime in. Many states (like North Carolina, where I live) allow .223/5.56 to be used in hunting deer and hogs. That is spreading because of the popularity of the AR-15 platform. (I know, the Elmer Fudds are having heart attacks, seeing them scary "black rifles" in the woods!)
I certainly have no aversions to taking a "black rifle" into the woods and greatly regret the fact that like handguns, it's illegal to shoot an AR in Canada outside of an approved shooting range. A shooting buddy of mine who used to 4x4 in the backcountry to the north of me used to carry one with a full 30 round mag as his bear defence gun back in the days that it was okay to do that. People up here still hunt coyotes with similar .223 rifles that aren't subject to such restrictions (i.e. Mini-14, various CZ-58 variants that can even take AR mags, the SU-16F, etc) and I'd do the same if I had such a rifle.
The issue with .223 for deer sized game rests on a couple of factors. For starters it just breaks 1000 foot pounds of pressure at 100 yards (900 ft lbs being the recommended minimum for deer), whereas even the 7.62x39 by comparison is just under 900 foot pounds at 200 yards. The .223 bullets are also lighter, and if you were to use the typical varmint load (SP or HP), the bullet could very well break apart before getting really deep into the animal. Even at 100 yards, you would probably want to refrain from anything less than a clear broadside shot where you can be reasonably assured that the shot can penetrate to the vitals. Up here the woods can be pretty thick, so if I know that I'll be shooting under 100 yards I'd sooner pick a good slug gun or heavier centerfire before choosing up a .223 that could easily be deflected by foliage
If you live in an area where bigger game like bears, elk, or moose abound, the .223 is hardly desirable for them while the .308 would handle all of those animals without difficulty. In some parts of North America, white-tailed deer vary considerably in average size, with smaller subspecies generally found further to the south, which may further dictate the appropriatness of .223 in some locales. As for hogs, I'd probably not see a problem using .223 on one from a tree stand where I can carefully take my shot, but it would hardly be my favourite choice for a drive!
.223 might be cheaper for target shooting, and it would be cool to have an AR if you already don't but even on the range or in the battlefield, the .308 has the advantage at longer ranges when wind's an issue. That's why the M14 is still used as the Designated Marksman Rifle by units in Afghanistan. A .308 bolt gun or M1A will give you lots of fun shooting and take you to your limits.....