300 yards is about the limit if your using a conventional scope. If you have a scope with reference lines for hold over or you will dial up, either will reach out ALOT farther then 300 yards. You drive a 300 gr bullet to 2600 fps or more, you still have plenty of punch at longer ranges.
That said, the game you will be hunting generally are not long range targets. Moose can be but often there is no real need. Bear, very few guides will let you shoot very far at one just to assure proper shot placement.
The 375 H&H is often eggagerated as far as its recoil energy. In a conventional sporter weight rifle in this chambering, recoil is suprisingly comfortable and most anyone can become accustom to it quickly if they have basic shooting skills.
The 375 RUM is a different story. For some reason, Remington decided to put these into a very lightweight, for caliber, rifleand as such, recoil can be classed as severe by many that are not used to shooting such a chambering.
A muzzle brake
will eliminate this problem on any rifle but with an increase in muzzle blast.
Simply comes down to this, the 375 RUM will offer you more velocity and energy but at a cost of recoil and muzzle blast. No moose or big bear will survive being hit with a properly designed 300 gr 375 bullet put through the vitals from either chambering.
If your a bit recoil shy and do not like muzzle brakes
, 375 H&H would be my choice. If you can handle more recoil or are not opposed to a muzzle brake and the issues it brings, I would opt for the RUM.
I have shot and used both and both can be extremely accuracy and consistant.