The biggest difference between mud tires and all-terrain tires is where you can use them. All-terrain tires have the added benefit of working well on and off the road. Meanwhile, mud tires are more suited to off-roading only.
They're a bit more robust than all-terrain tires. But, they can be uncomfortable to use on the road. They're loud and don't usually perform well on slick surfaces after a rain. Mud tires are great for dedicated off-road vehicles. However, they can be more of a hindrance on multi-purpose rides.