Simply stated the corsette came first, as either a full-coverage undergarment (covering the breasts) or an under-bust garment (worn with a shift or chemise).
|Underbust with chemise|
The typical early corsette was structured with whale-bone stays and a lace-up back for tightening. They were designed to push all the delightfully squishy female parts into perfect placement, pushing up the bust, cinching the waist and even rolling the hips back for more booty. At the turn of the century, the whale-bone stays gave way to steel (think underwire on steroids). Historical fact: In 1917, the US War Industries Board asked women to stop wearing corsettes to free up metal resources for war production. The movement was a success, and as a result freed up some 28,000 tons of metal (how’s that for coming together?! That much metal built the Board exactly two new battleships)! The one truly defining difference between corsettes and bustiers is the lace-up back, for a truly "custom" fit.
Bustiers are the comfortable cousin (if any lingerie can truly be called "comfortable"). They are not adjustable, and can also be full-coverage or underbust. It will have boning, be form-fitting, but really not as molding as the corsette. Bustiers tend to be more affordable than true corsettes. Both bustiers and corsettes come with attached or detachable garters for stockings.
Any more questions? Shall I cover the plethora of choices in bras and panties?