They are almost the same, but not quite.
Scrubbing means to rub something to clean it - usually with a brush.
Scouring means to clean, or to wear down a surface by abrasion.
Scrubbing describes an action, but if brushing isn't enough, the surface will remain dirty.
Scouring implies that the methods are enough for the job, and they can be anything. For example, many rocks in the Sahara desert are scoured by the wind. That means they actually are worn smooth, and it is because of the wind.
Scouring a person or animal would be very cruel.