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It is obvious that shut and close are very similar in meaning. However, while learning English as non-native (a long time ago) the teacher underlined that the two words are not identical in meaning.

I was unable to find a "clear" explanation about this difference yet - so I am still confused.

I found:

• the definitions of the words;


• how the words are similar;

I did not find: how are these words different.

Edit: Based on the answers and on the comments, is it safe to assume that there is no "general" rule about when to use "shut" and "close"? And that one must learn the "correct" verb for each situation separately?

The words have a sense which is completely identical, as seen in "shut/close the door". However, close can be used in another way, too.

If we accelerate, we can close the gap.

Essentially, close can be used to represent openings becoming smaller or objects getting closer. Usually based on the phrase close the gap. It can also be used metaphorically, as in "close the gender pay gap". You cannot use shut for this sense. Likewise, shut is used in certain instances where close would not be acceptable:

I've had enough of your lip, just shut up now!

(Shut up meaning "stop talking"; lip in this instance meaning impudent speech. Shut up is idiomatic in most if not all varieties of English; lip in this instance may be British-specific. Such instances of shut might be considered set phrases.)

Another thing to be wary of is that close has a homograph (same spelling, different pronunciation) meaning "near".

You can shut the window

You can close the door

You can close a highway

But you cannot shut the store

The verb close is an antonym for open. Therefore, we can say things like:

• Why did you close (or shut) the window? Please open it again.


• Please shut (or close) the door. Otherwise, the cat might get out.

In addition to doors, windows, briefcases, and cages, though, we can also open and close things like highways, runways, and restaurants. In this sense of the word, shut does not function like an antonym – at least, not in American English (the following examples don’t seem to hold true in British English, based on some informative comments beneath my original answer):

• Oak Street is closed due to flooding. We will have to find another way there.
  
  (NOT: Oak Street is shut...)


• The restaurant is closed. Let’s go find someplace else to eat.
  
  (NOT: The restaurant is shut...)

As an aside, the phrasal verb shut down can be used in a similar way as closed, but that often connotes a longer period of inaccessibility:

• That restaurant was shut down three months ago.

Put another way, a pub may have a closing time, but it will not have a shutting time.

It would be hard to come up with a comprehensive list. You just have to pay attention to what meanings each as, and see which ones are had by only one. If you want to replace one with the other, you have to check whether the meaning survives. One are to be careful in is phrasal verbs. "shut up" is generally used to mean "be quiet", while "close up", if it's used at all, is used for things like filling in holes. "shut out" means to exclude, while "close out" is used in liquidation sales. "shut in" refers to agoraphobics, "close in" means to get closer.

Also, the past tense and past participle of "shut" are just "shut", while the past tense of "close" is "closed". So you shouldn't say "The door has been shutted" or "The door has been close".

From other answers we can see that English speakers do not always agree about when these words are used, or should be used, even if we only consider their use in isolation rather than in expressions such as shut up and close down. I'm hard pressed to explain why I would more usually say

the airport was shut

rather than

the airport was closed

though I would view both as formally correct. This may be due to my Yorkshire origins: shut sounds stronger in an Yorkshire accent.

If we go back to origins of words I note this etymology site has

Old English scyttan "to put (a bolt) in place so as to fasten a door or gate, bolt, shut to; discharge, pay off," from West Germanic *skutjan (source also of Old Frisian schetta, Middle Dutch schutten "to shut, shut up, obstruct")

and this sense of not only closing, but fixing closed accords with my personal feeling of shut as being more forceful than close.