There are 4 large case .375" cartridges that provide excellent performance and for which brass is readily available; these are the .375 Ruger, the .375 H&H, the .375 Ultra, and the .378 Weatherby. The cost of .378 Weatherby brass is intimidating if not staggering, particularly if you want a few hundred, while the cost of Ruger, H&H, and Ultra brass is comparable and easier to swallow. The performance of the .375 Ruger and the .375 H&H is for all practical purposes the same and is bested by the .375 Ultra by 100 fps or so. The complaint that a magnum length action isn't as stiff and therefore not as accurate as a shorter action sounds just reasonable enough to be believed, but it doesn't work out that way in real life. My Brno 602 with it's 20" Ron Smith barrel chambered for the .375 Ultra shot like a varmint rifle; in all honesty though, while fine accuracy benefits one's confidence, its completely unnecessary when the typical .375 targets and ranges are considered.
The greatest advantage a long cartridge has is when long bullets heavier than 300 grs are loaded, but the majority of .375 owners will probably argue there's no such thing. When I tried loading 380 gr Rhinos in the .375 Ruger, the velocity was disappointing, about 2000 fps IIRC, the H&H will drive this bullet at 2250, and my Ultra will drive it at nearly 2400. At the other end of the spectrum, my current 22" .375 Ultra drives a 270 gr bullet at 3000 fps, which produces a pleasingly flat trajectory for those occasional longish shots, but the Ruger and the H&H aren't far enough behind that the difference matters very much.
Shooting big case .375s is not cheap, and if cost is an overriding consideration, you might be better off considering a .375 Scovill, which is a 9.3X62 opened up to .375. Of course this is a custom proposition, and considering the difference in cost between .366 and .375 bullets of similar weight, the proven journeyman performance of the 9.3X62 makes it the obvious choice, what with moderately priced factory rifles being commonly available. An additional bonus in favor of the 9.3X62, is that the rifle probably weighs 3 pounds less than a rifle chambered for a big case .375, a fact that can well be appreciated after a long day on the trail, whether you're breaking trail through snow drifts in Canada or slogging along a sand river in Africa. In those countries which mandate a minimum of .375 for dangerous game, the 9.3X62 is the sole exception to the rule. Many professional hunters have reported that any difference in the apparent effect on game between the 9.3X62 and the .375 H&H is, for the most part, indistinguishable.