As Ashley says, 'shut' can sound more abrupt and rude.
For example, after the dentist has examined your teeth, he or she might tell you that you can close your mouth. But if you are angry with someone and you want them to be quiet, you would tell them to 'shut ' their mouth.
However, in most cases they are interchangeable. But, as is always the case with synonyms, they are not interchangeable when it comes to phrasal verbs, set expressions and idioms.