jQuery used to ONLY have the callback functions for success and error and complete.
Then, they decided to support promises with the jqXHR object and that’s when they added .done(), .fail(), .always(), etc… in the spirit of the promise API. These new methods serve much the same purpose as the callbacks but in a different form. You can use whichever API style works better for your coding style.
As people get more and more familiar with promises and as more and more async operations use that concept, I suspect that more and more people will move to the promise API over time, but in the meantime jQuery supports both.
The .success() method has been deprecated in favor of the common promise object method names.
From the jQuery doc, you can see how various promise methods relate to the callback types:
jqXHR.done(function( data, textStatus, jqXHR ) {}); An alternative
construct to the success callback option, the .done() method replaces
the deprecated jqXHR.success() method. Refer to deferred.done() for
implementation details.jqXHR.fail(function( jqXHR, textStatus, errorThrown ) {}); An
alternative construct to the error callback option, the .fail() method
replaces the deprecated .error() method. Refer to deferred.fail() for
implementation details.jqXHR.always(function( data|jqXHR, textStatus, jqXHR|errorThrown ) {
}); An alternative construct to the complete callback option, the
.always() method replaces the deprecated .complete() method.In response to a successful request, the function’s arguments are the
same as those of .done(): data, textStatus, and the jqXHR object. For
failed requests the arguments are the same as those of .fail(): the
jqXHR object, textStatus, and errorThrown. Refer to deferred.always()
for implementation details.jqXHR.then(function( data, textStatus, jqXHR ) {}, function( jqXHR,
textStatus, errorThrown ) {}); Incorporates the functionality of the
.done() and .fail() methods, allowing (as of jQuery 1.8) the
underlying Promise to be manipulated. Refer to deferred.then() for
implementation details.
If you want to code in a way that is more compliant with the ES6 Promises standard, then of these four options you would only use .then().