Pick the caliber you like, either one will do the job.
I've killed elk with 120g Partitions out of a 25-06, I've used factory loads in a 30-06, I currently haul a 270WSM around the mountain and have killed a bunch of elk in the last 6 years with it. My hunting partner keeps borrowing my 300 WSM ( he has more sons than he has guns to go around) He has killed 5 elk the last 3 years with it shooting 168g TSX..
I shot a spike bull with the 270WSM using a 140 TSX, He was facing me at 230 yards. The bullet entered the base of his neck, follow the spine down and exited by his tail. Thats end for end of a spike bull. Just how much pentration does a person need.
Either of the guns you suggest will work. Pick the one you are comfortable using. I liked the 25-06 because I would go hunt Jack rabbits with it all winter. I could touch off 40 rounds in an afternoon, from all position. It was great practice. If I was using one of the big magnums, I'm afraid I might be flinching after that many.
Mt brother shoots a 300 wby and I shoot a STW. Not a lot of difference besides the 180 grain option for the 300. I shoot a 140 grain Swift A-frame thru the STW and it does a nice job on anything within 400 yards. I will say that as recoil goes the 300 does have a little more kick on the receiving end but that should be expected with a heavier bullet. Both of these guns are Model 70 winchester Sporter LT's and they are nice accurate pieces. In the end, ammo is expensive, recoil is definitely there but terminal ballistics are hard to match in either class 7mm or 300 calibers. Your opinion is what matters and good luck.
Over the years I have seen several elk and deer hides left in the woods by hunters and I have to wonder why they do this? I fully understand and agree about getting the hide off the animal as soon as possible to cool the meat, but why not pack out the hide with you and use it? As far as I know there are no state laws that require you to take the hide home, but to me why waste such a beautiful part of the animal? Some might think they have no use for the hide or it costs too much to tan....