Well, there are many options as far as ammunition is concerned for big game hunting. But when you are looking for ammo that is really powerful and has the ability to take down the target with one shot from a long distance then you don’t look to see beyond this three ammo. Apart from being powerful, all these three ammo are fairly accurate in their segment. Along with accuracy their projectile is really flat hence carrying a lot of energy to long distances.

But having said that all these three are suited to big game hunting from a long distance there is some massive difference between these three. Basically, they were designed to accomplish different goals and hence built with certain specific features. Though there is little overlap in their use especially still all three are distinctively different.

Actually, the magnum era started when Winchester introduced a line of belted magnum cartridges using a .375 H&H modified case. .338 win mag was for hunters in North America for large or tough games. After .338 win mag which was not an immediate success, Winchester introduced .300 Win Mag Ammo in response to 7mm Remington magnum which was an immediate success and remains a very popular choice for big game hunting.
.338 Lapua magnum was initially designed for a long-range sniper shot to be used for military purposes. After achieving success in military use it slowly made its way into the hunting community. It's one of the best when it comes to carrying relatively more power through long ranges.

While .300 Wing Mag uses a slightly longer case with a shoulder that sits .156″ further forward than on the .338 Win Mag case, both cartridges have the same 25-degree shoulder angle and the same overall length of 3.34″. Both cartridges are also belted magnums with the same .532″ rim diameter and the same SAAMI maximum average pressure of 64,000psi. On the other hand, the .338 Lapua is significantly longer than both Winchester Magnum cartridges. Lapua cartridge also uses a larger, .588″ rim diameter. Not surprisingly, it has significantly more case capacity than both the .300 and .338 Win Mag. It also has a higher maximum SAAMI pressure of 65,000psi. However, while the .300 and .338 Win Mag will both fit in a standard/long action rifle, the .338 Lapua Magnum requires a much longer rifle action.

Finally, the .300 Win Mag utilizes .308″ bullets while the .338 Win Mag and .338 Lapua both shoot larger .338″ bullets.
The trajectory of all three is very much the same up to 500 yards. .338 Lapua magnum has the highest recoil among the three. Coming to the comparison between the three Lapua is the most powerful but has the highest recoil where has .300 win mag has the least kinetic energy among the three but is still power full. Lastly, the .338 win mag is in between the two it is more powerful than the .300 win mag but has lesser recoil than 338 Lapua magnum ammo.

Currently, all these three cartridges are manufactured by leading manufacturers like Hornady ammo, federal, Winchester etc. out of them Hornady Lapua magnum is quite popular among members of the hunting community..... https://www.foundryoutdoors.com/collections/300-winchester-magnum-ammo