ORIGINAL: Rammer
I have a Browning A-Bolt Stalker in 7mm Rem Mag, and one of my hunting buddies has a Browning A-Bolt Stalker in 300 Win Mag. I sight in his gun for him every year, and I can tell you the 300 has more of a sharper "hit" to the shoulder, and the 7mm has more of a steady "push" in recoil. Both have the same muzzle blast, and the same muzzle flip.
I've never shot a Remington bigger than 30-06, so I can't help ya out much. Other than the Remington 30-06 was alot easier on my shoulder than a Savage, or Ruger M77 30-06.
Thats a decent example. I'd say as well its a lot like shooting an over/under shotgun (not as far as the amount of kick is concerned), but you can tell a difference between the bottom barrel and the top barrel... its more muzzle rise than anything... and its not a heck of a lot, but you can tell if you pay attention to it. Same as shooting a light weight semi-auto shotgun, then switching to a pump gun in the same gauge. Its just a bit different.