Obsolete. What a funny word. How is it defined?
Yes, Uncle Sam declared both the 30-06 and the 1911 to be obsolete, then proceeded to stockpile huge amounts of .30-06 ammo and weapons under the Emergency War Supplies program (which are still stockpiled), and recently dig out the .45s assigned to the mothball fleet ships for re-issue. So we can gather that obsolete is a political term that changes as the political winds and needs of the moment change. Flintlocks are considered as obsolete as the bow & arrow, but let the anti gun legislators decide to ban primers and reloading apparatus, and many of us would hunt and target shoot with those obsolete tools. If you use it, it isn't obsolete to you. [Note: using obsolete gear when the other guy doesn't may be hazardous.]
Neither the .308 nor the .30-06 is a practical short (100 yards or less) range Whitetail deer cartridge. I say this, having taken deer with both, because the bullet just zips on through and wastes the bulk of the energy on whatever is behind the deer. This phenomena doesn't usually stop until 300 yards or so. From 300 to 700 yards or so the difference between the two cartridges is so small as to not be worth mentioning. Not too many of us hunt game beyond 400 yards as a routine shot. Sure mountain goats, mule deer, etc., but a lot more game is taken much closer. When you do routinely shoot game at ranges way, way, out there odds are you shelved your .308 or your 30-06 a few weeks or months ago and went with (at least) a 30-06 improved, a .300 or a 7mm Magnum or something like that. So this brings us to the guns themselves as the big difference.
You have the 30-06 military surplus weapon designs primarily bolt action in nature. I have never met anyone hunting Bambi with an M-1 Garand. Nor have I heard of anyone hunting with one of military style 1918 B.A.R. clones. Probably for the same reason I don't use my M1-A NM as a Bambi rifle. It is just too durn heavy to be a good woods gun.
The sporterized 03s, and 1917s and Mauser 98s are much more practical. There are the civilian bolt action rifles such as the Winchester 70 or the Remington 700 or the Ruger and they are just as practical as a woods gun. Then we have the pumps and the lever actions. All of those design types are available in either caliber. [However there are limitations as to bullet weight and chamber pressure when dealing with the lever and pump actions.] As someone pointed out, you can probably shave 5 to 10 ounces off the rifle weight by using a .308 in a short action bolt rifle. Of course recoil energy is a function of rifle weight so what you save in carrying, comes back to you when you shoot. I have a 6 pound .308, so I know all about that. I can shoot the military 1917 in 30-06 all day and not notice the recoil, but 10 rounds from the 6 pound .308 and I have shot enough today. They both do ragged 5 shot 1 hole groups from the bench at 100 yards, so the accuracy is comparable at practical hunting ranges.
Availability of ammo. I visited Walmart today looking for some 32 acp or 35 Remington ammo. Found neither one there. Didn't even see any .45 acp there. I did however find lots of 30-06 and .308 and .40 S&W (also 1 lone box of 9mm) along with overstocked amounts of 12 gauge and .22 LR. My conclusion is both .308 and .30-06 ammo are here to stay for at least another 2 years or so. Neither is obsolete yet.
Bullet weight is where the .30-06 starts stepping out in front of the .308. Depending on the load it has 5 - 10% more power. Supposedly you can still special order 220 grain 30-06 ammo. I have never seen that offered in .308. Very useful for plinking at moose, but a little light for Kodiak bear or charging lions at close range. Likewise the twist rate of many rifles is probably such that the long range accuracy of ultra heavy bullet loads is dubious at best. [I'll let someone else research that issue.] Light weight 110 grain bullets in both calibers have been hard to find lately. However I still have some accelerator rounds for both calibers, and I don't see any difference between those on my groundhogs.
My conclusion is the difference between 30-06 and .308 is a difference that makes no difference from any practical viewpoint.