A bustier is a sleeveless top that spans the bust to the waist (sometimes touching the hips or the top area of the butt) and it is worn either as lingerie or as outer clothing. It is form fitting and it may sometimes have garters attached to it. Modern bustiers now have thin lacy straps but most bustiers are like tube tops.
The corset was developed more than a millennium ago. It is a close fitted boned undergarment that shapes a woman’s torso. It can either extend from above or below the bust to the hips, giving that appropriate binding effect to have a flatter waistline.
As you can see from the definition above, the two are more likely to be called as twins. They look the same and they practically do the same thing. But in the interest of settling the debate, one huge difference between the two is that the corset came first in terms of origin while the bustier came trailing decades later. A more prominent difference is that a corset does not always cover the breasts while a bustier always does. A corset that covers the chest area is called an over-bust corset.