Downhill control is the biggest one. Even on a steep downhill grade on pavement a manual affords you enough control that brakes are reserved for the idiots in front of you. Ever seen a big rig with an automatic? You also get a lower first gear ratio which is great for technical trails, and rocks. Many of them have a slightly higher final drive ratio, paired with the more efficient connection to the engine results in better mileage and less drive line stress while towing. Your reverse gear is also usually geared lower affording better control while reversing.
Now I will say this it's possible to use an automatic for all this stuff just make sure you've got a good transmission cooler, and use synthetic trans oil. Autos don't like heat and in 4low they can build up quite a bit of it. Especially in deep mud where they really do shine as they are well sealed as long as you run the breather up.
So I guess when we all say a manual gives you more control it's probably 80% in regards to more efficient engine braking, reducing the amount your on the brakes a lot. The other 20% is picking your gear ahead of time and being able to go through better than with an automatic which may have to downshift twice as it looses momentum from the the first downshift. Restarts on a steep hill may not be fun in a manual but they are a lot easier with the safety feature that makes you push the clutch in first disabled, or you do it enough to know how to work 3 pedals at once.

I've got some friends who've never been able to get that and now they both have autos in their JK's.