A region of a city requires capitalization if the direction word is part of its name, so the West End of London, North and South Philadelphia, and East New York all take capitals because they are official names of neighborhoods. New York also has the East Side and the West Side, which are recognized formal names, but there might be another city where "the west side of town" may have a different, specific proper name.
The appropriate rules come from http://www.write.com/2014/05/14/capitalization-of-cardinal-and-intermediate-directions/:
Cardinal and Intermediate Directions
Cardinal directions are the four basic points on a compass—you
remember— that primitive thing that used to help people find their way
before the invention of GPS. The four cardinal directions are north,
east, south and west. Intermediate directions, such as southwest,
northeast and so on, make up the other compass points.
Cardinal and intermediate directions do not need capitalization when
they are simply used to describe directions:
- Writers will learn proper capitalization of directions when pigs fly north.
- The disgruntled birds canceled their plans to fly south for the winter, fearing collisions with flying pigs.
The same goes for directional terms ending in –ern and –ward. These
words also require no capitalization when used merely to describe
- Munchkin farmers raised aerial pigs in western Kansas.
- The old witch melted southward and disappeared, much to the delight of the elated munchkins.
Proper Nouns and Regions
When a directional term is part of a proper noun or used to describe
an entire region, it requires capitalization:
- Wake me up when the pigs reach the North Pole.
- Winged swine filled the Southern sky that night.
- No one on the West Coast eats bacon, but he’s from the South.