I've shot deer with both the .243 and the .25-06. The first is good and the second is great. In California, the .243 is fine for dear, but when you get outside this state, the .243 goes marginal. The .25-06 drops them where they stand.
I've shot two antelope in CA. The 7mm Mag. broke both shoulders and the animal ran 60-70 yards. The second was with my 25-06 and the bullet entered about an inch behind the shoulder at about the same distance. The animal dropped like gravity had been reinvented where it stood.
I've not hunted with the .257 Weatherby, but I know the 25-06 pushes the powder capacity to bore diameter ratio and the Weatherby has got to be way beyond.
My newest .25 caliber is the .257 Roberts Improved, and its as good as the .25-06, but that requires alot more work.
You really can't go wrong in either choice. You can always load the .25-06 down to the .243 but you can't load the .243 to the .25-06.