The phrase "do the washing-up" typically refers to washing the dishes or cleaning the dirty utensils and plates used in cooking and eating. It is primarily used in British English and is equivalent to the American English phrase "do the dishes."
On the other hand, the term "do the laundry" refers to washing clothes, bedding, towels, and other items typically done in a washing machine or by hand. This task involves cleaning garments rather than dishes.
So, to clarify, "do the washing-up" means "wash the dishes," while "do the laundry" means "wash clothes and other fabrics."