If it were me, I would be necking up 300RUM cases and trimming to suit, the reason is this, even though unlikely, necking 375 -338 MAY cause the brass in the neck to thicken. You'll be pushing the shoulder back with either case, so just choose one or the other.
To answer your other question. Remington HAD to make the 338RUM shorter because the ogive of most 338 bullets would be inside the case neck when seated to the proper length for functioning if it were left at 2.850", so instead of making the neck shorter they shortened the entire case, exactly the same as Winchester did with the 300 and 338.
The difference in length will make no discernible difference in performance.