Is there a difference between chicken broth made with breast vs. leg meat and why?
It's also a matter of palate and preference. Some simply don't care for dark meat at all, much less boiled on the bone. For some it has a bitter, or even metallic aftertaste. 

Breasts are lean cuts with a rather mild flavor. They don't contain nearly as much fat, and are far more susceptible to overcooking. Thus, a broth made with breast meat is going to be much milder in flavor (especially if you intend to eat the poached breast). 

The more a muscle is actually used during the life of the animal, the more flavor it's generally going to have. This is because of enzymes and acids that maintained the muscle while the chicken was living and then started to break down the meat once the chicken was slaughtered. You also have bones, which have marrow, which holds quite a bit of flavor. Thighs are one of the fattiest parts of a chicken. So yes, the flavor is a bit more intense. 

I don't use either in my stock, I buy our chickens whole and break them down to individual cuts (the price of the sum of the parts of a chicken sold individually is usually higher than the price of a whole chicken). This gives me the carcas, which is mostly skin, scraps, feet, neck, back, bones and cartilage. It creates something with a jello-like consistency due to the gelatin in the bones.

I boil then simmer them for several hours, skim any impurities off the top, chill the stock and then remove the layer of fat that rose to the surface and solidified. This becomes the base of my broth since it doesn't have the aftertaste some want to avoid. I then poach cubed breast meat if I want chicken soup, or use it as a braising liquid if I want to stew some legs and thighs.

So if you know everyone eating loves dark meat, use dark meat, it's delicious! If you want to have a sort of neutral base that you can go either way with, just use the carcass, including as a base for something made with just white meat.