A region is a large or extended land area that has some geographical, cultural or language characteristics that set it apart from others. It can also be indefinitely extended, as with 'the regions of the cosmos.'
A region can also extend across countries. The region of the Alps covers several countries.
An area is usually smaller. It can refer to a specific area by some definite border or an indefinite land space.
The two can mean the same thing, though. I can talk about a hilly region or area.