There are many reasons to have a 300 Win Mag over a 308 but there's also plenty of reasons to stick with the 308, esp in a lighter weight rifle. I'll let others cover the pros and cons.
On the issue of recoil: First, I'm built about like you from your pictures and I don't like punishing recoil at all. I think the 700ADL will weigh about 8.5-9 lbs with a scope. The 300 Win Mag recoils roughly about 50% more than a 308, 270 or 7mm Rem Mag. I have all three of those and the main 300 I have experience with is a very light weight custom job. It kicks a lot! If I went from a 308 to that 300, I'd probably never shoot it enough to get truly proficient enough with it to chose it over the 308.
For some reason, unlike almost every other American rifleperson, I have an aversion to the 30 caliber. I prefer the 25-06 and 270 to the 30-06 and the 243, 260 and 7mm-08 to the 308. There's nothing wrong with killing an animal just a little bit deader but I just don't believe that the 30 is the optimal caliber. I prefer the 7mm Rem to the 300 Win as for the popular magnums. In rifles of equal weights and comparing the standard bullet weights and velocities, the 300 kicks about 50 more. This is a noticable difference. I can't imagine any animals within 500 miles of you or me that th 300 will drop instantly but the 7mm will let get away with equal shot placements.
Ammo costs, either factory or reloads, are going to be about equal. If you need penetration, the 7mm should generally edge out the 300 and heavy bullets are available. if you want minimal meat damage, no problem. If you want expansion, you can certainly get that. If you want solids, you can have those as well. Bullets are available to the handloader from 100 Gr to 180 Gr and to the factory shooter from at least 140 to 175 Gr and that covers pretty much everything in North America and a whole lot in Africa. It is noticably less recoil, a bit flatter trajectory and a well proven game-taking ability.
Between the 7 and the 300, I might feel a bit more comfortable with the 300 if specifically going out to look for big bears. Actually, I'd feel even more comfortable with a 45-70 but you get the point.
Edit: I think I just talked myself out of selling mine.