Although between does express some kind of relationship involving two of something, we must be careful to define the word's meaning in a way that would not reject a sentence like this:
There is a treaty between the five great powers.
Looking at the raw number of entities involved in the utterance is not a good way to decide whether or not to use between. The two-ness can be a very abstract thing, not just a physical thing like two parked cars.
You can walk between two parked cars.
You and your cousin can settle a dispute between the two of you.
The ball can pass between the goalposts.
You might have to complete an assignment between now and next
You might even hear a person reprimand a group of six middle-school boys who have been found doing something stupid and dangerous:
You nitwits, you don't have an ounce of common sense between the lot of you.
or hear something like this:
We all wanted to chip in to buy him a gift for coaching the peewee basketball team. Between the eight of us, we were able to get him a box of his favorite cigars.
What does between mean in those sentences?
There the idea is one of mutual contribution, each person contributing towards a common "pot", in the first example, the pot of common sense, in the second, the pot of money needed to buy the coach a gift. It is the relationship between the person and the "pot" that is expressed by between.
among expresses the underlying idea of in-the-midst-of-many.
He walked among the crowd.
Is there a doctor among you? One of the golfers in our party needs medical
Among the reasons for the obesity epidemic in the United States
is the ubiquity of high-fructose corn syrup.