I hunt with 6.5x55mm and .284 Win, exclusively. In many ways, these are long action versions of the two cartridges that you're between. I have hunted with a .243 Win, however the 6.5- and 7-mm cartridges are far better all-purpose hunting tools, pragmatically. Seems that the .243 is eliminated from the pool anyway.
I live in northeastern PA, and hunt in our forests and mountains for whitetail; additionally, my wife and I have a hobby farm, where we raise heritage breeds of waterfowl, and we deal with predators and pests, from coyotes to rats and critters of all sizes in between. Due to the forested conditions, shots are rarely more than 50-yds unless I'm scanning a field, where the shots can be out to 700-yds.
If I had to choose from the 6.5 or the 7 for all-purpose in my own use, then the 6.5 would be the better choice. If I had to pick one of the two for my wife to hunt with, then again it would be the 6.5. I load with 100-grain through 160-grain projectiles, subsonic and supersonic depending on the purpose. For the 7-mm, there are plenty of great projectiles available for hunting anything from predators through moose, but I can do everything that the 7 is capable of with the 6.5, just 100-yds closer.
If you're not loading your own rounds, then availability of factory-loaded hunting ammo is a factor. In terms of recoil, I have owned 7x57mm rifles of moderate weight (8 - 9-lbs.), and the recoil is a bit more than a hot Swede round with a projectile of equivalent mass. The 7x57 is a pretty fair comparison with the 7mm-08. My .284 Win is long action, and is more of a light 7mm Magnum. My Swede loads are considerably faster than a factory 6.5CM, and I still find the recoil light on a 9-lb. rifle with a 127-gr LRX.
Hope this is helpful.