The army experimented with the 338 Lapua, but decided on the 300 WM.
The 338 offers slightly better perfromance, but lots of negatives. The Army already had 308 rifles built on long action Remington 700's that could be converted much cheaper than buying a new system.
Right now there is a mix of both 308 and 300 WM rifles. Rifles are gradually being converted to 300 WM and put into service. I could see keeping some of both. The 308 is actually a very good choice in situations where extreme precision is required at more moderate ranges, 200 yards or so. That is why it is still prefered by LE snipers. The 300 offers a huge advantage at 1,000 yards +.
but idk how it stacks up against a .300 in a similar weight rifle.
Using this website
From a 7.5 lb rifle
300 WM 180 @3100 fps 33 ft lbs recoil
30-06 180@ 2800 fps 22.5 ft lbs recoil
308 165 @2760 fps 17.2 ft lbs recoil
338 Lapua 250 @2880 56 ft lbs recoil
A 7.5 lb 338 isn't a realistic weight, nor is a 300 win mag
From a 10 lb rifle
338 Lapua 42 ft lbs recoil
300 WM 25 ft lbs recoil
Of course most 308 sniper rifles are going to be 10 lbs +. I just happend to already have that data from a 7.5 lb hunting rifle.