I agree with the above. There is an onomatopoetic quality to them, though, which makes "shut" sound a bit more abrupt.

Both the sentences are right, at least in US English. When referring to things like windows and doors, there is no difference I can see. In some expressions, there is a difference. Examples When talking about finishing something and moving on, you "close the book on it." Shut wouldn't work. When one team doesn't score in a game, there are "shut out." Close doesn't work. If you make a gap smaller, you "close the gap." Shut doesn't work. When you end a discussion, you either "close the meeting" or "shut off further discussion." These are just the ones that come to mind.

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.