Size
The American Foxhound is much larger: about 30 kg (65 lb) as an adult, against 18 kg (40 lb) for the Beagle.
Grooming
The American Foxhound needs more grooming, brushing a few times a week, where the Beagle needs a brush once a week.
Exercise
Plan for plenty of exercise with the American Foxhound, and a moderate amount with the Beagle.
Temperament
They differ most on intelligence (American Foxhound: easy-going, simple routines; Beagle: learns readily, likes games) and on handler demand (American Foxhound: needs an experienced owner; Beagle: needs steady training).
Alike
Both are rated the same on independence: decides for itself.