You scrub something when you use something stiff and scratchy like a scrubbing brush to clean it with. When you scrub something you do it vigorously, not gently.
A tub is a large container with an open top. It is wider than it is deep. Large tubs used to be used for doing washing in, but today tubs are usually used to put things in, or for growing plants in.
Tub is also a short word for bathtub or hot tub.