In most common usages, they are equivalent. In other words, in any type of situation where you move something so that some type of hole or gap is filled, either close or shut can be used with no difference in meaning. For example:
Shut your eyes. / Close your eyes.
Shut the door. / Close the door.
Shut your mouth. / Close your mouth.
Shut the window. / Close the window.
Shut the refrigerator. / Close the refrigerator.
Here are a few special situations where one is preferred over the other (at least in American English):
We say you close a bank account. You don't shut a bank account.
We say a store closes at a certain time. We don't say it shuts at a certain time (not sure about UK English).
We say "Shut up!" We don't say "Close up!"
The price of a stock closes at a certain amount. It doesn't shut at a certain amount.
You close a deal. You don't shut a deal.
Hope this helps.