I've had both, a .375 H&H and a .375 Ultra mag. The ultra mag is like the H&H on steroids. I sold the H&H and the .375 Ultra mag is one of my favorite rifles.
I've taken my .375 ultra mag to Africa twice, taking 18 animals from Steinbok to Cape Buffalo. Most were one shot kills at ranges from 30 yds to 348 yds. The first trip I was shooting 300 gr Barnes TSX bullets at 2840 fps. The second trip I shot 270 gr TSX bullets at 3043 fps (that's 5553 ft-lbs of energy!). Both bullets shoot 3-shot sub moa groups. That's accurate enough and enough energy for any bear or moose on the planet.
My .375 ultra mag is a model 700 Remington that out of the box was very unpleasant to shoot. I restocked it in a laminated stock (that fits me!) and is pillar and glass bedded, has a Limbsaver recoil pad, and a mechanical recoil reducer in the stock. I also have a KDF muzzle brake
on the barrel. 95+ Grains of powder is LOUD with or without a muzzle brake. Unloaded and without a sling my .375 ultra mag weighs 9 1/2 lbs. It's now comfortable to shoot, even from a prone position.
Someone mentioned the availability of ammo. .375 H&H ammo has been around a long time and is more widely available than ammo for the ultra mag. You will also pay a premimum price for .375 ultra mag ammo. If you handload, you can taylor the ammo to your rifle, and the ultra mag ammo will only cost about 5 cents more for each round than handloaded .375 H&H ammo.
If I ever hunt Cape Buffalo again or Asian Water Buffalo or Brown bear, my rifle of choice will be my .375 Ultra mag with 270 gr Barnes TSX bullets.