I have Ruger #1's in both .375 H&H and .416 Rigby. I like the Rigby a lot and loaded with .300gr Barnes X-bullets at over 2800fps, or the Speer 350gr Mag-Tip loaded just a little slower, it will really reach out there and put a heavy duty hurt on anything that walks in North America, and most of the rest of the world. And loaded with 400gr premium bullets it will very effectively flatten anything in Africa (if you do your part).
But the choice and availability of bullets for reloading is not nearly as good as the .375, and the .416 components (especially brass) are more expensive. And if you buy factory loaded ammo (if you can find it), expect to pay $100 and up for a box of 20 rounds, whereas .375 ammo is available in the $20 to $30 price range and is widely available.
For hunting in grizzly or brown bear country, I wouldn't hesitate to use the .375. With the 210gr Barnes X or 235gr Barnes XLC, you can get velocity up over 2900fps (possibly even above 3000fps). But I had a friend that hunted exclusively with a .375 H&H Ruger loaded with Speer 285gr Grand Slams loaded to around 2650 to 2700fps, who took quite a few elk and moose and a few big bears with that load.
The .375 will be plenty of gun for what you say you want to do, and it's a lot easier on the shoulder and the wallet than the .416, which means you can tolerate and afford to practice more with the .375.