First, I have no experience with a Kuvasz so there is nothing I know first hand with that breed. But I grew up on a farm with working dogs, we had predator issues and talked with other farmers with the same problems and had a chance to see AS and GPs at work as well as what their owners thought of them. Here's what I can relate from my experience and what their owners told of their performance.
1. Both the AS and GP are great guard dogs. Seriously--I heard nothing but good things from their owners (one guy had sheep, another cattle). We had some BC's and my dad had grown up with sheep and cattle out west so we had a professional curiousity about how those dogs did.
Both are truly outstanding at what they do but they're a bit different in personality though. The GP is more of a nocturnal dog. If you're dealing with predators (like coyotes) than that's probably your dog. But if you've got livestock that might be outside during the day but in the stable or corral at nighttime, a nocturnal dog isn't much of a fit.
Additionally, I was always told (and from the interaction I saw), the Pyrenees is a more social dog that will fit into the family better, interact with children better if you've got those. But if not, or if the primary concern is solely a good protector of your lifestock, I'd go with the AS. The AS tends to take the attitude of: anything I've been introduced to (owner, his kids, the lifestock, other pets) is in the circle, they're okay. Everything else outside the circle better stay away.