My thoughts...
The Jeep weighs less, and likely has more ground clearance overall making it seem easier to drive through deeper snow. You also mentioned that it has mud tires, maybe they're wider than the 4R allowing you to disperse more of the lighter vehicle and 'float' over it.
Regardless, 13" of wet snow means that you were plowing with your under carriage, which isn't easy when the snow is heavy and you're going uphill. Snow conditions will dictate a lot of what you can and cannot do, not necessarily the depth alone. I have been through 18" light powder in mine with zero issues, and 6" - 10" of the heavy stuff where it took more effort to get through even getting stuck. Your best chance at traction in in those conditions would be 4LO, A-TRAC on, and
momentum. Momentum is your best friend, and if you get stuck don't force it, just back up, gain momentum and go again. Using the locker isn't a bad idea either, you just have to remember to turn it off once you're moving, but arguably in these conditions you'll get just as much traction by using A-TRAC alone.
Here is a great article explaining all of the 4Runner's 4wd gizmos and how to use them:
5th gen off-road features
Here's another article explaining winter travel tips:
4x4 winter travel tips
Best of luck!