Big Model 70, 375 H&H fan here, however...
I just ordered an H-S Precision PHR in 375 Ruger at SHOT. Theoretically, the more modern cartridge design of the Ruger variant is slightly better and it does have the potential to get you another 100 fps, though I doubt that translates to any real field advantage.
The Ruger is further complicated by the availability of factory ammo. I've seen two of the Rugers in the possession of African professional hunters and both have sung its praises, ammo is on the shelf over there, albeit not as widely as the Holland & Holland. The way ammo is now handled when traveling makes it very unlikely yours would be lost anyway.
The real issue is bullets. The only factory ammo I am aware of is Hornady, I'm not a big fan of conventionally constructed bullets on Dangerous game. Therefore firing the 300 Barnes, Trophy bonded, Nosler homogenous solid, etc. is strictly a handloaders proposition. If you're not a handloader, or willing to sit behind a 375 for dozens of rounds working up a load at the bench, the H&H is a better choice because of the huge selection of bullets on the shelf in factory loaded form.
I have three 375 HH's, a Ruger #1, M70 and the Kimber Caprivi. One of the PHs I hunt with has the CZ. All are superbly accurate, no doubt accuracy inherent in the time-tested round when chambered in a good barrel. Nothing to choose from here.
I've taken the M70 on four hunts now and done-in perhaps 20 animals with it, from steenbok thru giraffe, buffalo and hippo. Next up is elephant. It sports a new 26" 5r Bartlein, VAIS muzzle breaked barrel so it gets 2650 out of factory 300 Barnes, pounds them in under an inch and kicks like an '06. With the McMillan Hunter stock I was surprised to see it accepted four in the box, one in the pipe makes five. I believe it's the perfect, do-all African gun and highly recommend these particular upgrades.
It's my go-to African gun. My PH says that if I should show up without it, he'll be happy to wait while I go back and get it. That perhaps says it all.
The Number One Tropical is of course a single shot and kicks like a mule so it hasn't seen any overseas field use. It'll cloverleaf 260 and 300 Noslers all day long and would be a superb large-plains rifle.
The Kimber is a bit heavier, holds an extra round in the box (4) over a factory Winchester and comes up quickly and naturally. It's not quite as accurate as my custom 70 yet. Aesthetically, it's a stunner, looks like it just belongs in Africa. As a Model 70 'clone' I believe it'd make a great safari rifle but I haven't taken it yet.
I see the CZ as more of a professional hunters rifle than clients due mainly to its weight, and a lot of them carry one. Their weight comes from the fact that they have a magnum Mauser receiver, not necessary on a 375 but is useful in larger bores. In a 416 Rigby they'd be very high on my go-to list. They are widely distributed over there, relatively inexpensive and that is a big factor in their popularity. Some guys that own them swear by them.
The CZ stock design seems to take Americans a bit of getting used to, as does its cock-on-close functioning, and they almost all need some gunsmithing or a lot of use to smooth-up the actions as others have mentioned. I keep hearing about their stocks cracking so I'd be sure to bed it as aftermarket fiberglass stocks hard onto come by for the CZ magnum. At some point I'd surely like to have one, top of the heap in big bores, but for the 375 I'll stick with the 70, 700, 77 and the like until I feel the need for a big-bore bolt gun.
Will the new H-S Precision 375 Ruger supplant my M70 Dragonslayer (as my PH calls it) once I get it? That remains to be seen...