Recoil difference is noticable. Not as noticable as difference between an 06 and win mag, but near that.
I hope I don't offend too many, but one cartridge I loathe is the weatherby lines. The freebore is just more than I can take. I am talking almost 1/2".
You gaining a modest 100-150fps with a weatheryby. If I was going that route, I would just go ahead and get a 300RUM, which I have had 3. Recoil between RUM and weatherby as more simular than between the weatherby and win mag. At least with the RUM, you headspace on the shoulder. The RUM still has .2" freebore. Dont' dare get one without the freebore. I know a guy who insisted on touching the lands with 3.6" COL partition. The gunsmith did exactly what he said, and he ended up not being able to shoot factory ammo. And was lucky to get 3200fps with 180gr accubond without pressure signs.