Size
The Great Pyrenees is larger: about 65 kg (143 lb) as an adult, against 41 kg (90 lb) for the Bernese Mountain Dog.
Shedding
Their coats shed differently: the Bernese Mountain Dog sheds through the year, the Great Pyrenees moults most as the seasons change.
Exercise
Plan for a moderate amount of exercise with the Bernese Mountain Dog, and a modest amount with the Great Pyrenees.
Temperament
They differ most on independence (Bernese Mountain Dog: keen to work with you; Great Pyrenees: decides for itself) and on affection (Bernese Mountain Dog: a lap or leaning dog; Great Pyrenees: affectionate on its own terms).
Alike
Both are rated the same on prey drive: usually fine with small pets.