Here are the facts: a waistcoat is a garment that covers a man’s upper body and is sleeveless, while a vest is also a garment that covers a man’s upper body and is sleeveless. So, there really is no difference between a vest and a waistcoat.
Whether one term is used or not is also geographical. In England, for example, the garment is more commonly referred to as a waistcoat, while in America we tend to refer to it as a vest.
And to some, a vest is considered an informal garment and a waistcoat a more formal garment to be worn with a tie.
Bottom line: Either one is fine, though at HSS we prefer to use waistcoat. Informal or formal, here are a five ways to wear a waistcoat.