For those with power steering it is two fold. The variable steering is nice 16:1-13:1, instead of 17.5:1 lock to lock. The ratio is slower while near the center of the box, and faster the closer you get to lock out. This maintains highway speed drive-ability while lessening the amount of turns necessary to go from lock to lock(handy while maneuvering and low speed). So essentially it's the best of both worlds from a ratio stand point.
The H1 box also has larger/stronger internals than the CJ box, making it more durable when paired with larger tires etc.
My CJ had a box that came out of a Chevy Nova/Camaro, it was a 14:1 constant ratio box. This was a mistake by the PO-the box was a Saginaw 800(AKA 76 Box), but they made that same casting with 2 sets of internals(different ratios) one for Trucks(17.5:1) and one for cars-the number near the top plate differentiates the two(hard to see in junk-which is probably how it made it into the jeep). The Jeep was really twitchy and nerve racking to drive.
With the new H1 box, the Jeep is much more stable, and drives like a whole new vehicle. But like I said. I had the wrong box to begin with, for those with a proper box, the difference may not be as drastic dependent upon it's condition.