Left to right: .50 BMG with 800-grain bullet; .338 Lapua with 250-grain bullet; .300 Win. Mag. with 190-grain bullet; and .308 Winchester with 175-grain bullet.
Do you need more than a .308? That depends on what you plan on shooting and how far away you plan on shooting it.
The traditional sniper/tactical cartridge today is the .308 Winchester, due to its accuracy and ubiquity. The big problem with the .308 is that any bullet from it goes subsonic soon after 1,000 yards. Passing the trans-sonic zone has a bad effect on accuracy. To strike a heavier blow, something like a .300 Winchester Magnum gets suggested. Based on a bigger, belted case, it delivers heavier bullets at higher velocities. However, it only delays things a bit. The .308 boots a 175-grain bullet at 2,600 fps, the .300 launches a 190 at 2,950. However, you pay a lot in recoil to add only a couple of hundred yards of supersonic travel.
You want more, you have to get bigger. The big medium bore is the .338 Lapua Magnum, which fires a 300-grain bullet at 2,750 fps. It doesn't drop below the speed of sound this side of 1,800 yards. How does it do that? Simple, the ballistic coefficient (how well it slips through the air) of a 300-grain .338 diameter bullet is so markedly superior to any .308 diameter bullet as to be no contest. So a good shooter with a .338--assuming he knows how to read wind, mirage and distance--can get hits well outside of the effective range of a .308-equipped shooter.
The biggest normal cartridge is the .50 BMG. Until the Barrett family of rifles, long-range .50 BMG shooting was a stunt. The "Big Fifty" (not to be confused with the old .50 Sharps rifles) has so much going in its favor that it does not drop subsonic until it has passed 2,200 yards.
Downrange performance? The terminal effects of all these bullets are in direct proportion to bullet weight and velocity. If we use the extreme end of the .308--1,000 yards--we find its 175-grain bullet is just deciding to go subsonic, which is roughly 1,100 fps. Not the hammer of Thor, but not something I'd want to play tag with on opponents who might be shooting back.The .338 Lapua does much better, and at 1,100 yards its 300-grain bullet is delivering over 1,600 fps of terminal velocity to the target.
At 1,100 yards, the .50 BMG is still doing nearly 1,800 fps--and that with a 750-grain bullet. Ouch. Having fired a .50 BMG on 1,000-yard targets, I can tell you it makes a satisfying spray of gravel out of any rock smaller than a steamer trunk. Mild steel plates immediately get holes in them. What is "cover" to lesser cartridges is merely concealment to a .50 BMG.
However, you pay for what you get. The recoil of the .300 is noticeably more than a .308. The .338 is about all most shooters can fire and enjoy the experience. The .50 BMG requires a muzzle brake of some kind simply for the shooter to survive the experience. The larger cartridge size and recoil also require larger launching platforms. On the other hand, a .308 tactical rifle is basically a normal hunting rifle with a heavier barrel and sturdier stock. At 10 to 12 pounds it is like other rifles, and the same-size rifle can be had in .300 Winchester Magnum without too much extra bulk or weight.
A .338 Lapua such as the Sako TRG tips the scales at 11 pounds empty. Add a scope, sling and ammo and it can be as much as 15 pounds. A .50 BMG is almost off the scale. For instance, Barrett's M-82A1 is 31 pounds without scope, ammo, sling or accessories. And you need the weight and muzzle brake to shoot it.
Get as much as you need, but not more than you can carry or handle.