I've never owned the WSM, but it seems really popular and looks pretty cool. (I have tried it on the range, and it recoils about like a 7mm Remington Magnum with a 175 grain bullet.)
However, I have a fair amount of experience with a .300 Winchester Magnum over a number of different rifles. This is why I, personally, would stick with the .300 Winchester Magnum. My own personal experience counts for me, and I know the old-fashioned .300 Win. Mag. works. For example, I know to first load a 180-grain bullet with 73 grains of IMR 4831 when starting to work with a new rifle in .300 Winchester Magnum. If this load doesn't give me groups in the range of 1 m.o.a. or better for three shots, I know that something is wrong with the rifle, and I will start checking for problems.
I would not be in a similar situation with the .300 WSM because I don't know the magic "universal" loads for it.
Apart from that, I'd personally go with the rifle that actually feels the best in your hands, and comes up to your shoulder like it was a natural part of your body. If given the choice between a .308 that carried like a dream and fit me perfectly, and a .300 Magnum that felt like a club and bobbed around when I brought it to my shoulder, there would be NO choice in the matter. A hunting rifle must work for HUNTING. That means, it must be comfortable to carry and come up naturally, even when your muscles are aching and tired, and you are shivering cold.
My Sako .300 just points like it is a part of my arm. The shape of the old Sako Monte Carlo stock seems to be perfectly made for my face and the length of my neck. I can bring it to my shoulder with my eyes closed, and when I open them, it's pointing where I want it to be. That's the reason why I always want to grab it when I go hunting.
The fit of the rifle to your personal ergonomics is far more important than cartridge choice, in my opinion (within reason, of course -- we're not talking .223 for grizzly). Therefore, you should actually shoulder the rifles you are contemplating and choose the one that feels the best to you, and not worry about the minute differences between the .300 WM and .300 WSM, or 7mm Rem Mag (or .30-06), for that matter.