In the "olden days", like 90 years ago and earlier, people would get dressed up in fancy clothes and go to the "ball". That meant a big fancy formal dance in a large formal room with chandeliers and ladies in beautiful full gowns and men in suits. It was a type of a planned Dance that was meant for rich people, because poor people in the countryside had different types of dancing....probably in barns, with country musical instruments made out of wash tubs and fiddles they enjoyed.
When someone planned a "Ball" and invited people to come to the dance, they would have ballroom dancing. Just doing a dance did not mean you were doing a "ball". It is not a synonym.