I used to hunt with 243 when I was a kid, now my son has that rifle, I can tell you I took my share with both 7-08 and 243, I have seen the damage both rounds inflict and how fast both rounds drop an animal with well placed shots, the 7-08 ammo I load and use does a much better job than any 243 {I wasnt loading back then} I ever shot.
Also as far as the recoil, I don't feel a huge difference between the 2... If I were new to hunting and looking to get a new rifle in a white tail caliber for medium distance and tree hunting I would go with the 7-08, hands down a more capable round.
I use 139 gr sst with the 7-08, I took a 205lb 10 pointer in KS, my rifle is 0 at 300 and he was around 500 yards out, I had him in my binos calling and calling be he couldn't hear my call so I screamed a call at the top of my lungs, he perked up and looked over {I swear he looked right below me at the tree I was in}, sure enough he started walking in, I was watching him through the range finder now checking his distance, he kept looking like he was going to turn around, and my ride out of the woods was due anytime and would surely scare him away!!! I was stressed, it was also the last day of a 9 day hunt and I had a tag with no deer!!!
So I as soon as he came close to 300 I put the range finder down grabbed my scope and put the reticle on him, but now he was coming square at me, I tried to turn him with some loud claps but he was walking right at me, I decided to do a front hear shot {first and only I ever took} he dropped before I could get my eye back on him, I thought I missed him so I went to the binos, he was gone... When my ride came I drove over and there he was right where my rifle left him...