The difference between 'between' and 'among' is not - I repeat NOT - a simple question of whether you are talking about two or three items.
This very misleading advice is widespread among non-native teachers, particularly in China. As a result, millions and millions of learners and believe that it is correct to say, for example, "What is the difference among A, B C and D?'. This error comes up several times a day on italki.
The correct question should be 'What is the difference between A, B C and D?' . We use 'between' for ANY number if you are considering them as individual items. Or, for example, you could say to two friends, 'Let's keep this a secret between the three of us'. As Rae says in her comment, 'between' is very precise - we know how many things or people are involved.
'Among' and 'amongst' (which has the same meaning) are much more vague. For example, you would say 'I found this old photo among my books and papers'. Here you imagine a mixed jumble of things and you have no idea how many there are.
This page has plenty of explanations and examples:
http://blog.oxforddictionaries.com/2015/06/grammar-myths-among-or-between/