There are advantages that each has and the other doesn't.
Competitive rock crawling, climbing and maneuvering over boulders and tight places first gear is the only gear used anyway, so it's really not doing anything "automatic."
The advantages of an auto are no clutch to operate and the torque converter's multiplication of torque is nice. Plus it makes things a little simpler - rear steer, steering brakes, lock/unlock - it's getting pretty "busy" for a competitor without the clutch and shifter to think about.
But then, there are times when being able to feather the clutch is nice too. "Rocking" it to get loose is much more effective and useful with a clutch.
And of course there's the overheating/cooling problem with an auto, the vulnerability of the pan getting punctured, and more weight.
But then lots of folks have trouble operating 3 pedals with 2 feet.
For hill climbing it's nice to have the auto backshift if needed, it's quicker than a manual (and no danger of missing a shift,) but it can also cause you to break traction if it shifts at the wrong time. Unless you have the right valve body in the auto, it will backshift when it wants, not when you want it to.
For desert racing when you are using several gears, up and down constantly an auto really helps with the fatigue factor, just like in stop and go traffic.
For general Jeeping, not competition, I prefer the better control of the manual trans.
Like I said, I have 2 with manuals, and 3 with autos. My more serious rigs are manuals. But now that I have foot problems I may need to convert.
Notice all sorts of racing has mostly gone to autos - dragsters are computer controlled shift points - no time for the driver to make the decision to shift. Formula cars are also mostly autos now too.
Still, it's mainly a matter of personal choice for most Jeepers.