Between is very specific. It can talk about time or space. You could also use "in the middle of" or "on either side of."
Space: "The school is between the church and the store." ("The church and the store are on either side of the school.")
Time: "She has a lot of time between her first and second class of the day." ("She has a lot of free time in the middle of her classes.")
Among is more general, and usually talks about a person or object being included in a group. Synonyms could be "surrounded by" or "included in."
"The flower was among the weeds." ("The flower was surrounded by the weeds.")
"The girl was among the first group of students to go on a field trip." ("The girl was included in the first group of students.")