You develop more control over the movement with the box squat. You're always lowering your body to the exact same point, so you groove the movement better than if you do squats the traditional way--deeper with lighter weights, less deep with heavier weights.
(Side note: I remember watching a guy break his squat
PR about five or six times in a row at a gym. He was amazed that he could just keep lifting more and more weight. His secret? By his fifth or sixth
PR, he was barely going a quarter of the way down. But he thought he was the biggest stud who ever walked the earth by the time he was done.)
The bar can be high or low. In competition, obviously, you'd place it lower, but some competitive lifters train with it higher, since it's harder that way.
One thing I didn't know until I watched Dave Tate at Westside is that the object of a powerlifting squat is to move the bar straight up and down. You don't want to bend forward at all. That's why they hold the bar so low and spread their feet so wide.
The first time I tried Westside-style box squats, my knees were killing me, and I thought I wouldn't be able to do them. But within a week I could do them with heavier weights and no knee pain at all.
It just takes some getting used to.
Also, I think the fact you actually stop and sit on the box develops your core strength in a different way. You can't let those muscles go slack in any kind of squat, but there's something about the discipline of holding the position, and then standing again, that feels different.