The difference
A wrist corsage sits on an elastic or ribbon band. It is the most common choice for prom because it does not damage the dress and stays put while dancing. It is also easy to take off for dinner or photos.
A pin-on corsage attaches to the dress or jacket, usually on the left side near the shoulder. It is traditional for mothers and grandmothers at weddings and looks polished with a structured dress or suit. It can pull on thin fabric, so it is not ideal for delicate dresses.
Magnetic backings are an option some florists offer for pin-on corsages, which avoids holes in fabric. Ask the florist what they can do.