Both words join things together. But between is used where two things are mentioned and among is used for more than two.
We say “Between you and me, between one and two o’clock, between May and June, between State Street and Dearborn.”
We say “Among the members of Congress,” as there are many of them. We also say “Among the baseball teams in the American League; among thieves (providing a clue that there are more than two); among the several widows who live on my street.”
Both words can be used in the same paragraph, which could go something like this:
Mark McGwire, Sammy Sosa, and Barry Bonds were among those who applied to coach my son’s Little League team; and the final choice was between Sosa and Bonds. Sosa got the job.