Rub and scrub are similar. Both usually when using the hands. When I clean the shower in the bathroom there are some hard to clean stains. I have to use a lot of force for that so I'd use the word scrub here. It also for me has the connotation of cleaning. Rub could be used in cleaning but for example to remove a stain or mark. You'd say to rub on it. Mop is used with a long-handled thing called a mop. If you spill a lot of water on the floor you will mop it up with a mop