As others said, if you don't reload, the 270 Win can be found more places and easier then the 25-06, plus you get heavier bullets for the 270 Win.
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A bit off topic since the OP says he doesn't reload BUT:
If a major manufacturer actually made some match-grade .257 bullets we could actually compare apples to apples.
Berger does. I shoot both, but I have a 257 Rbts. instead of the 25-06. I don't think anyone disputes the 25-06 shoots the same bullet a bit faster then the Roberts.
But I don't see, under normal conditions the 243 out preforming the 257. First, in factory rifles both normally come with a 1:10 twist. Meaning you are limiting the weight of the Bullets. The 243 would be about 90 gr. The 257 about 115 Gr.
Both my rifles are Winchesters and have a 1:10 twist barrel. I shoot Bergers in both, 87 gr for the 243 and 115 for the 257 Roberts. The 243 starts out at 3000 fps and the Roberts at 2800 fps. The Roberts goes subsonic at 1500 yards, the 243 at 1450, the energy difference at those ranges is 300 for the 257 and 222 for the 243.
In reality they are pretty dern close. The 257 will have a bit of a wind advantage over the 243,
But again, my comparisons are with the 257 Roberts, you're going to have more of an advantage of the 25-06 simple because its faster then the Roberts.
I also shoot the 270 Win (also in a 1:10 twist Model 70). Again using the Berger bullet, its got them both beat. Starting at 2750 the 150 gr Berger remains supersonic to 1650 yards and bucks the wind better and more remaining energy.
Of course we wont be hunting at those ranges, but this does give you an idea between the three.
My Bergers are the hunting VLD in all three rifles. Berger engineers recommend the remaining velocity of 1800 FPS for their hunting bullet to work properly. That would be 800 yards for the 270, 750 for the 257 and 800 for the 243, so for deer size animals they are pretty near the same.
You move on to elk, then the 270 would hold the advantage.
When all said and done, at normal hunting ranges, lets say 300 yards, there isn't a nickel's worth of difference on deer/antelope size animals.
Except those Berger's are nasty, I'll stick to my 257 or 243 for deer/antelope and use the 270 for elk.
The OP mentioned deer, both his choices would work great, the 270 would hold the advantage when it comes to finding factory ammo.
As to the 300 Win. I don't care how big and tough you are, the recoil will get to you. There is a reason long range shooters are deserting the 300s for the 6.5 non-magnums. I gave up my 300 win for 1000 yard matches for the 308 Win a long time ago. It will beat you to death.