The surface area of a shape is always the same, it does not matter how you find it, just as it doesn't matter if you use a pocket calculator, a computer, or a pencil and paper method to find 183+123. The answer will always be the same.
Fundamentally, there is some quantity you are measuring, and these are just different methods to measure it. We don't do these methods for fun, we do them because we want to know some quantity--the surface area, in your case. Don't understand the surface area as the answer to this addition problem, but instead understand it as the surface area (ie, how much cardboard you'd need to make this shape in real life) and try to understand the addition problems as ways to count this area.