I would have to say I think they are both going to be well on the side of adequate when properly loaded. As Craig noted I don't believe the nonsense argument that 375 is better than 30, but 338 isn't, but you have to draw the line somewhere. I would personally pick 338 even knowing its going to make my life more difficult and expensive than it needs to be, but I couldn't argue in good faith that 300 isn't good enough.
Lets say for sake of argument that we want to pick one bullet for each to go to alaska to hunt. Lets pretend we know that we are going to need to shoot a grizzly bear who is not thrilled with our presence at 50 yards, and the next day we are going to need to shoot a trophy caribou at 500 yards. Are we going to bring one load for both of those hunts or are we going to have two loads and prepare ahead of time to be able to switch from one to the other?
What bullets would you want to have in the 300 win mag when the grizzly bear is close enough to smell? Looking at what's available in 200+ grain 308 bullets I would say a 200 or 220 grain nosler partition, 200 grain Swift A frame, 200 grain Norma Orynx, 220 grain Sierra pro hunter, 200 and 210 grain barnes LRX. Those are all probably okay but I think I would be nervous about the barnes because I have not had great experiences with copper bullets. Now the next day we want to shoot our caribou at 500 yards. Are any of the previously mentioned bullets a good choice for that? Most of them have terrible ballistic coefficients. There are lots of heavy pointy 308 bullets that would be fine for the caribou, but the option that I would want to use for both are pretty slim. The only 2 bullets I see over 200 grains, are of tough construction, and have a good BC are the 210 grain barnes and maybe the 210 grain accubond long range though I think that will be pretty lightly constructed for the impact velocity at close range. The nolser 200 grain partition or 200 A frame would be decentb but not a great BC for long range shooting. Most of them are either soft frangible bullet like an ELDx or Berger, or they are on the lightish side and will be going way over 3000 fps at the muzzle, which is not what I really want for the bear. I think a two bullet plan would probably be best for the 300 win mag, which means having two different zero's. Not a huge deal, but something to consider.
Looking at 225+ 338, I think the Nosler accubond in 250, (or 265 if you could get them) become reasonable choices for bear because the muzzle velocity will be a lot lower than 300, Nosler Partition in 225 or 250, Barnes 225, 250, and 265 grain options, the 225 Hornady CX, Swift A frame in 225, 250, and 275gr, and maybe the Sciracco in 210gr, Speer grand slam 250, Norma Orynx in 230. Which of those are good choices for both? I think the accubonds would be excellent. The Sciracco would be good but a little on the light side for the bear. The barnes and Hornady CX would be okay if you are okay with copper bullets. The partitions, A frames, or Speer grand slam would be fine but not the greatest BC so I would have to check to see if they would stay in their expansion window out to 500 yards.
So in my assessment if you want a single load to do it all I think 338 has a little more flexibility. If you are okay with having two specialized loads and two scope zero's they both have workable options. I went through this thought experiment myself and came up with 375 ruger. If the 265 grain 338 accubonds were actually available and I could buy 500+ of them I think I would probably go with the 338.