I had a 338/06 built 5 or 6 years ago with the intentions of making an elk hunting trip every year or so. Retirement hasn't gone well for me so the hunts haven't happened.
I feel the bigger caliber is beneficial in some cases compared to an equal weight but smaller diameter. One of the places I've hunted has limited range but heavy cover so a 300 Win mag's range is not an asset. I felt the .338 cal bullet would deliver the energy somewhat better at the shorter range than the .308 cal.
All my ammo is handloaded and most of it is using reformed 35 Whelen brass although I did score a good deal on a small lot of WBY 338/06 factory brass that no one else recognized for what it was.