The 375 is adequate no doubt. If AND ONLY IF you can shoot the 416 as well, it will of course hit harder and give you more margin on buffalo, assuming it isn't too heavy to carry, the optics are up to the task, all else being equal, etc, etc. Very few people are heard to complain at the end of a successful hunt that they wish they had put a smaller hole in the buffalos chest. But as a client, I will never shoot enough buffalo that the greater impact of a 416 vs a 375 has any statistically significant chance of making a difference in outcome. I'll be lucky to shoot 10 buffalo in my lifetime and for as rarely as things go wrong and as modest the improved performance of the bigger round is, it almost assuredly won't make a difference for me. No doubt a PH will have enough chances for things to go wrong and a 458 Lott or a 470 or whatnot has enough increased performance on buffalo that it is a better choice for him.
As far as what to choose if plains game are involved, I am going to disagree with the vast majority who say XX rifle for dangerous game and YY (smaller) rifle for plains game, and I know my opinion is well outside the mainstream. Based on personal experience, I don't see what advantage a smaller rifle offers. I shoot my 404 every bit as well as my 30-06 (which may say more to my lack of ability with the 30-06 than my ability with the 404!) and while the 30-06 is and always will be adequate for plains game, the 404 hits harder and drops animals faster. If you shoot the 416 as accurately as you shoot the 375, the 375 isn't going to do a better job on smaller animals, it is just something else to carry. Either is more than you need but there is no more dead than dead. If it were just a plains game hunt, you might want to bring a lighter rifle that is easier to carry all day, but no amount of weight savings on the plains game rifle is worth traveling with 2 rifles for me. I agree 100% with Red Leg that one gun is enough to carry.
I'll also disagree regarding the 416 being preferable for elephant: as the client, you will be taking a single aimed brain shot on a specific single animal. It matters not at all whether you use a 375 solid or a 416 solid, it only matters whether you place the shot properly. If things go sideways, a big stopping rifle would be welcome but such a rifle is only going to hinder you in doing your job as the client. Remember much of what is written about elephant hunting is written by ivory hunters or game department cullers. Those are very different situations than that faced by the client on a modern elephant hunt.
All of that said, if I had started with my 375HH R8, I doubt I would ever look to buy another rifle. My 30-06 is a little easier to carry. My 404 custom GMA arguably hits harder and inarguably has more classic beauty. The 375 R8 is easy enough to carry and hits hard enough. If I didn't have a lot of memories tied up in my other rifles, I'd be well-served to simplify my life and just use my R8 for everything.
In my experience, one is one and two is one too many) 
The .375 is not just a good compromise, it is to my mind, the best solution for such a multi-species hunt. I own a .470, 500-416, and .404. On a mixed bag hunt, I take one of the .375's and a quality 300 gr bullet. I do it because I believe it is the optimal choice for such a hunt.
