I have a Sako in .375 H&H and bought a used custom Brno in 375 Weatherby from a friend.
I find some advantages for the Weatherby over the H&H case: - you can use the 350grs bullets with good velocity - if your ammo pack gets lost, you can use also H&H ammo without problems - case stretching is not so pronounced as in the H&H case - for practise you can load "light" to H&H level
Disadvantages are: - price for cases and ammo - availability - need of a longer barrel (24") - of course, recoil
I have no experience with terminal performance of these two cartridges, I´m shooting both just for fun!