Hi Ariane,
Because of the history of Britain, we have a lot of 'double vocabulary' (sometimes even three or more words that are very similar). Most of the time, when we do have this double vocabulary, there is a word from the Romance languages (like French, Latin, etc.) and one from the German language (like German, Danish, Norwegian, etc.). That's the case here. 'Close' (as a verb) is from Old French, and 'shut' is Germanic. In this particular case, there is little difference between them, but 'close' is slightly more formal than 'shut'.
As a few people have mentioned, there are particular collocations (words that usually come together) where you really need to use 'close' and others where you need to use 'shut'. You just have to learn these, there are no rules. Most of the time they are fairly interchangeable.