The most common use of between is when something is in the middle of two things or two groups of things.
We use between to refer to two things which are clearly separated.
He sat in between Allen and Melissa. [=He sat in the middle of Allen and Melissa.]
Mongolia is in between China and Russia. [=China is on one side of Mongolia and Russia is on the other side of Mongolia, and Mongolia is in the middle.]
I will be at work between 8 am and 5 pm. [=8 am to 5 pm is the range of time you will be at work; you will be there for the time in the middle of those two times.]
The train runs between New York and Boston. [=The train travels from one city to the other, and the two cities are stops at the ends of a route.]
Besides, some nouns are often followed by between: connection, difference, distinction, link, relationship
Is there a connection between his arrival and her disappearance?
There’s a strong link between clean drinking water and reduced infant mortality.
The most common verbs often followed by between are: choose, differentiate, distinguish, divide:
[talking about judging a competition] I didn’t want to have to choose between the singers.
Her money was divided between her son and her daughter.
The most common use for among is when something is in or with a group of a few, several, or many things. Among is used when talking about people or things that are not distinct and are viewed as a group:
We use among to suggest a sense of being a part of or surrounded by or included in something else. It is typically followed by a plural noun phrase:
I sat among the other students. [=There were other students all around and I sat within the group.]
There wasn’t much unity among the council members.
He dropped the penny among the flowers. [=He dropped the penny somewhere in the flower bed.]
Sidney has several dancers among her friends. [=Within Sidney's group of friends, several of them are dancers, but the rest are not dancers.]
She wanted to be among friends.
Among his books, we found some rare first editions.
Paula always wanted to go swimming among dolphins.
In the phrases "among others" and "among other things", among means ‘as well as’:
Her parents, among others, were worried about her travelling alone.
Among other things, I still have to pack.