"What are the differences? "
A region is an area, it could be part of a country or part of the world ,which not necessarily has fixed boundaries , yet has definable characteristics .
A district ( smaller scale than a region) is an area inside a country or a city ,that is distinct, because of a particular characteristic or a certain activity .
A shopping district
An area could be that part of a city, country or the world , which has physical boundaries . It has an extent and size .
A zone is also an area that has particular characteristic and usually is bound to certain restrictions.
war zone etc..
The distinguishing between those words might be a bit vague ,but surely with more practice you will be able to determine which is the right and most commonly used word in a certain context !