Size
The Great Pyrenees is larger on average: about 65 kg (143 lb) as an adult, against 50 kg (109 lb) for the Anatolian Shepherd Dog.
Grooming
The Great Pyrenees needs more grooming, daily brushing, where the Anatolian Shepherd Dog needs a brush once a week.
Exercise
Plan for a moderate amount of exercise with the Anatolian Shepherd Dog, and a modest amount with the Great Pyrenees.
Temperament
They differ most on affection (Anatolian Shepherd Dog: likes its own space; Great Pyrenees: affectionate on its own terms) and on prey drive (Anatolian Shepherd Dog: chases; needs managing; Great Pyrenees: usually fine with small pets).
Alike
Both are rated the same on independence: decides for itself.