Here is my line of reasoning.
Long range shooting/hunting is a game of reducing variability and/or the effect of numerous variables including, but not limited to (followed in parentheses by possible solutions to the problem - BTW this is an incomplete list):
Inherent weapon imprecision (buy a custom rig)
Inherent ammunition imprecision (reload with quality components, sort brass, weigh bullets and charges, etc. - the sky is the limit here)
Inherent shooter imprecision (trigger; weapon fit; practice)
Ranging imprecision (laser range finder; flatter trajectory round)
Wind (practice; higher BC heavy bullet going fast)
Changing field or weapon conditions (tempature, etc)
Terminal ballistics on the animal (bullet composition; energy)
Ideally, we would like to reduce all these variables to a minimum, or reduce their effects on accuracy and terminal performance by employing as many solutions as possible, since all of the inherent limitations have a magnified effect the farther you're shooting. But tradeoffs are of course involved.
This is the whole reason to spend $3500 on a custom rig, rather than $900 on a Factory Sendero, so we can improve fit and reduce group size to under 1/2 MOA. You're spending $$$ to reduce variability. A lot of $$$.
So for me, I want to reduce the impact of other variables by using a round with ballistic characteristics that are also optimal - basically the flattest shooting round that will have sufficient terminal performance to kill whatever I'm shooting at at the ranges I'll be shooting at.
Looking at reloader's nest and using the NF ballistics program, we see that it is reasonable to get a 208 AMAX to about 2900 FPS in a 300 WM or up to 3200 FPS in a RUM (both of these may be high estimates, you can play with the numbers all night if you wish). The difference in energy, wind correction, and drop at 1000 yards (at Sea Level and 40 degrees) are 340 foot pounds, .75 MOA and 5.25 MOA - not a lot but not peanuts either, and all in favor of 300 RUM. At 500 yards (perhaps a more realistic distance for most game shots) it's 550 foot pounds, .25 MOA wind and 2 MOA drop, all in favor of RUM.
So, suppose he ranges an elk at 809 yards and rounds down to 800 yards to dial in the MOA correction for both guns. Suppose also that he either ranged the grass in front of the elk, or the elk has been imperceptibly retreating as he's watching it, and it's true range is 839 yards at the time he squeezes the trigger. Suppose further that he measures a 7 MPH wind that is really an 8 MPH wind and that he has corrected for it in both guns by rounding down from 7 to 5 MPH; that he shoots 1/2 MOA groups at that distance; and that he holds dead on in the center of the kill zone. Here is how much he's going to be off from his point of aim in both guns (I'm ignoring other variables such as temperature and the like):
300 RUM: elevation 1.25 MOA (difference between ranged and actual distance), wind 1.25 MOA (1.25 MOA from rounding wind speed and inaccuracy in wind speed measurement, 0 MOA from ranging error). Thus, he is shooting a 4" group at 800 yards that is going to be off his point of aim by 10 inches low, and 10" either right or left. If the kill zone of an elk is 26" in diameter (assuming it's circular for simplicity; reference: Chuckhawks.com) his point of impact is going to be in the kill zone 75% of the time and the bullet impacts with 1900 foot pounds of energy, or the same energy that a 308 with a 168 grain bullet at 2600 fps at the muzzle has at 175 yards.
300 WM: the corresponding values are 1.5 MOA elevation (difference between ranged and actual distance), and 1.5 MOA wind (1.25 MOA from rounding wind speed and innacurate measurement, and 0.25 MOA from ranging error). With these values, he's now 12" low and 12" off with wind, and in that kill zone of radius 13" only about 25% of his shots are in there given a 4 inch group. The bullet impacts with 1400 foot pounds of energy, about the same as a 243 shooting a 105 grain at 3100 fps at the muzzle and impacting at 300 yards!
These differences will be less at closer ranges (because the trajectories overlap there) but they will be magnified at longer ranges (where the trajectories diverge). Obviously if this were a mule deer, it would have been a miss. Or if the ranging error didn't exist, they would have [almost] all been kill zone hits. But that's what we're fighting - VARIABLES - and it's generally best to reduce them (and their impact [pun intended]) as much as possible.
So, what is the price that is to be paid for this in the 300 RUM compared with the 300 WM, and what weight should be given to these tradeoffs? I will comment on the tradeoffs mentioned by other posters on this and another board.
1.) Recoil - I dismiss this based on the same kind of argument used above. While distance magnifies differences, muzzle brakes and suppressors divide differences. Using the recoil calculator on handloads.com, the recoil impulse of the RUM in lbs*sec is 4.57 and that of the WM is 4.01. Reduced by 50% with a brake the values are 2.28 and 2.0, for a 14% bigger recoil impulse from the RUM. Suffice it to say if you are willing to shoot a WM without a brake, the RUM with a brake is going to be a walk in the park. The recoil impulse of an 8 pound M1A shooting a 168 grain at 2600 is 2.67, so the RUM is going to kick less than that (all else equal).
2.) Cost of ammo if you reload - RUM brass is a bit more expensive and you're going to have about 10-15% reduction in powder charge with WM. I'm not going to do a formal analysis here, but I think we can dismiss this.
3.) Barrel burn out - well, if anybody had asked my buddy how many rounds he throws down range in a given year, they would have learned that it's probably less than 200 in all his high powered rifles combined, and I know because I do his reloading for him! So at the rate he's going, a RUM barrel is going to last him 5-10 years, and when the throat wears out, he can have it cut and rechambered (in 300WM if he thinks it wore out too fast!). He's spending $3500 for a rifle, he can spend $500 on maintenance every 5 years. I go through a $500 pair of motorcycle tires once a season, you gotta pay to play.
4.) Cost and availability of factory ammo: Now you might have something here. I'm betting that the Nosler factory 200 grain loaded Accubond will be the best factory bet for the 300 RUM, but it's $89 a box at midway right now. The same stuff is $69 a box for WM. So if you shoot 100 rounds a year, you save $100 on factory ammo between the two calibers. You have way more choices with the WM however, and the Federal Gold Medal with a 190 SMK might be a good long range round with this gun, if you weren't going to reload.
I don't think I'm missing any salient arguments about trade-offs between the two calibers. The RUM has slightly better ballistics and terminal performance (energy). But it costs a bit more to shoot and recoils a bit more even with a brake. If cost is such a deal breaker, then I find it difficult to justify the price of the custom rig over a factory Sendero. If recoil is a dealbreaker, either mount a brake or get a 243 and sneak up on them.
But if you want to reduce variability as much as you can, get the 300 RUM.
That's all I got.