South is a noun. Southern is an adjective. You can live in the south or live in the southern part of the country. See how south is a stand alone noun, and southern supports "part of the country".

They can both refer to direction. Example: He lives in the south. He lives in the southern part of the state. Southern can also be an adjective to describe something or someone that lives in the south. Example: She has a southern accent. She's a southern girl.

South (and also North, East and West) are precise definitions. For example, the border between South Korea and North Korea is very exact. South Carolina and North Carolina are two different states, with a line separating them. Southern, northern and so on usually describe a more vague geographical area. The term 'southern France' , for example, loosely refers to the area starting somewhere in the middle of the country and heading southwards. Different people may have different interpretations of where an area described as 'southern France' might begin. NB The terms Southern and Northern Hemisphere may be seen as an exception.