Interesting, thanks! Doesn’t an awd car also provide power to all four wheels? How does that work and what makes it worse than a transfer case in the eyes of the national park service?
So there are a few things that make a transfer case and most 4WD systems generally better for off-road. The biggest two are gearing and how durable/robust the systems are.
The ability to change the final drive gives incredibly low gearing to 4WD vehicles. This provides increased torque for traversing obstacles and better low speed control.
In the durability front, that is largely due to design considerations. AWD vehicles are typically designed for on road use, so the engineers built them with that in mind. This means lighter and weaker drive shafts, weaker steering racks, less robust suspension, and generally lighter duty pieces throughout the entire system. 4WD vehicles usually designed with heavier duty used in mind (even if the majority of people won’t use them that way). This means the entire drive train is built stronger and more capable of abuse.
This is a huge generalization though. You could build a purpose built AWD off road vehicle with low gearing, and locking differentials if you wanted. And there are plenty of 4WD vehicles that wouldn’t survive even moderately difficult trails. So the NPS probably made the rule because generally 4WD is more capable for that use case, and it would be difficult to make an all exhaustive list of what vehicles or components would give you the capability to navigate the trail.
Four wheel drive is “dumb” in that it simply sends equal power to all four wheels. But if one wheel loses traction, then a quarter of your available power is wasted as that wheel just spins (this is a gross oversimplification). All wheel drive, by contrast, operates like four wheel drive but uses a fancy differential instead of a transfer case. If the system detects that one wheel loses traction, it can selectively direct power to the other wheels instead of wasting it. It can also lock the center differential so that it behaves like a transfer case if you want. I don’t know the specifics of the park service’s rules, but an all wheel drive vehicle should do just as well if not better than a four wheel drive one.