These comparisons are hazardous, as I usually end up buying the ones I don't have just to see for myself. But not in this case. I'll admit to being a .338 WM fanboy. Mine is not in the back of the safe, and I'm on the second loading of the 500 new cases I started with. First, a little comparison from Hornady's handy ballistic calculator. I have tested loads for the 200 AB and 250 SGK. I'll compare to Nosler's book load for .300 WM 200 AB. All 200 yd zero.
For 200's, there's no comparison. Nosler's .308 BC is 0.560, their .338 BC is 0.414. They start about the same speed. Nosler gets 2,972 from the .300 WM and a little over 3,000 with the .338. I, however, get 2,964 from my 24" Savage 116. That BC kills the .338 at long range. At 500, it drops 41.9" and holds 1,663 ft-lb. The sleek .308 drops 37.1" (not a big difference) and holds 2,123 ft-lb (big difference).
Now, my favorite 250 SGK. My Re-19 loads go 2,700 out of my rifle, pretty much in line with the manuals. Recently, I tried VV N560 and got 2,800 without going off the reservation. But I'll use the former for comparison. With a BC of 0.563, the GameKing drops 45.9" at 500 and holds 2,139 ft-lb. (OK, my faster load does 42.3 and 2,325. My new go to.)
Clear as mud, right? Oh, both the .308 200 Accubond and the .338 250 GameKing have outstanding sectional density. The answer to the OP's question is "neither". Either one will kill an elk at 500, or well beyond that if you can hit boiler room. Of course, so will a 7 mm RM or a 30-06 or... The .338 does have an advantage, I my opinion, at closer range. Big bullet at a little lower velocity isn't as likely to damage as much meat.
I bought my rifle intending to hunt Alaska, and I still hope to get there. Meanwhile, I've learned to shoot her (she does kick) and that's what I'll pack out west.
I'll close with some factoids. When I lived in SE Idaho, the .300 WM seemed to be the rifle of choice, not the only one by any means, but very popular. I carried a 7 Mag at the time and did well with it. But my elk hero carried a .338 WM. Dad was coming out to hunt, and I needed a rifle (He later gave me the 7 Mag). So I asked my hero. He spent 15 minutes on the virtues of the .338 WM, then said, "Get a 30-06." I did. Still good advice. Me, I like my medium bore magnum knock 'em dead, but it ain't for everyone. And 500 yards? When you get to my age, the coyotes will have eaten it before I get over there.