That's a good question and would benefit from the input of a true native speaker. Meanwhile, here is my understanding:

"À l'extérieur" refers to the exterior of something, while "dehors" is "outside of". There are cases where both would be OK but others, where you can't replace one by the other.

"L'extérieur de ta bouche est sale." -- You could not use "dehors" in this context to refer to the perimeter/outside of your mouth.
"C'est dehors la question!" -- It is out of the question! (Here you couldn't use à l'extérieur)

"L'extérieur" refers to a specific place, whereas "dehors" stands more for the outside in general or in a less literal way.

"Tu vis en dehors de la ville." -- You live out of town. (In a general sense)
"Tu vis à l'extérieur de la ville." -- You live on the "outside" of the town (in a more specific sense).

I hope that helps, -- Chris (not a native speaker).

Well.....yes and no. If you take geographical as meaning a well defined place as opposed to a more general notion of "not inside" then it is actually a pretty decent description. But it's hard to hang their difference on one word, geographical or not. :)

-- Chris.