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!

Rolf