Husky is a size for (fat) children's clothing, and suffers from the same overuse as a euphemism as stout does (as demonstrated in the thread to which cuchuflete linked). In both cases, the emphasis is on big, rather than strong.
My feeling is that stout almost always implies fatness. Brawny (which I think has a bit of a stupid connotation - not much between the ears, so to speak), stocky if they are small in stature, well-built, hefty (I would use with an object rather than a person) are all possibles.
"Husky" to me implies some size and strength, as well as more body fat than a fashion model. I don't use "well-built" that way -- but, then, I don't use it at all, so perhaps for others it's synonymous with "husky."
If you are an overweight adult man, you shop in the Big and Tall department. If you are an overweight little boy, your mother buys your clothes from the Husky department. Many boys have been traumatized by the shame of wearing Sears Husky jeans to school with Husky written visibly on the label.
Do we use 'husky' for anything but dogs and voices? I had only the vaguest idea what it might mean as a personal description, a euphemism for a slobbish bear-like man with a very bad beard with nasty looking bits stuck in it. They mainly live in the Alaskan wilderness.
In AE "husky" is an attempt by clothing manufacturers to avoid the word "fat" or "big" when labelling their clothes for fat and obese boys. I have never heard it being used outside that context for "fat". It's an attempt to neutralise the concept of fatness. People avoid saying "fat" here. It's a very emotive word.
I've never heard anyone call anyone else "husky". But I have only 10 years experience of living in the USA and mostly in Maine and New York City.
It's not so much a neutral word as a euphemism.