North, South, East, West, can be used as adjectives, but so can Northern, Southern, Eastern, Western. What rules, if any, govern which is appropriate when?
North can also be used as a noun: one of the points of the compass is called north, and "the north" is frequently used to refer to geographic regions. Northern is always an adjective (notwithstanding its occasional use as a proper noun, as in "I attended college at Northern").
If you had to choose one of them to use as an adjective, you would probably want to use northern. North as an adjective is generally limited to proper nouns or idiomatic/traditional usage: the North Pole, the North Star, a north wind, the north face of the mountain, the north end of town.
I observe that most uses of north as an adjective corresponds to a (presumedly) static location: north pole, north face, north side, a direct reference to the north as an orientation or a place.
Northern seems to be used as a direction, a movement relative to something else, a comparative. I note that northern is quite old and is attested as soon as the 12th, it could have an additional literary glow.
In the example given by phenry, Northern Illinois University, the northern is clearly used to distinguish this institution from other ones, using the relative situation as a badge. This strategy seems to have succeeded.
To me north can be used as an adjective but only in restricted situations, mostly conventional, meteorological.
It seems that "Northern" is used to name a certain region and not a specific place. Thus "Northern Illinois" and "Northern Ireland", whereas North Africa is the name of the country and not a region.
The use of specific direction as North, South, etc. designates a direction as in "I'm going South for the winter", to get to this place, you drive East for five miles, etc.