A powder room and a half bath are really the same thing.
A powder room is considered a half bath which in old days the user would go into the room and powder their wig or their nose.
But now we don't usually call it a powder room but instead usually just refer to it as a half bath but if a home you're looking at says it has a powder room they are basically saying it has a half bath.
A powder room or half bathroom is usually located on the first floor of a multi-story home, making it easily accessible to guests and family members who don't want to travel to an upstairs bathroom.
A half bathroom typically only has a toilet and sink and no shower or bathtub and is mainly used for applying makeup or using the toilet and washing your hands.