In You may not need jQuery (Maybe you don't need jQuery) two cases are distinguished, the use in the application itself and the one of including jQuery as a library requirement. I leave the case of using jQuery as a library requirement out of the scope of this question because otherwise it would be too broad. This question is focused on the use of jQuery in the application itself.
The reference suggests using jQuery whenever possible, however there are questions on Stack Overflow that ask for answers using pure JavaScript.
Today here on Stack Overflow in Spanish there is a question that currently has two answers, one with jQuery and the other with pure JavaScript. This question is not about saying which of the two answers is better, but about better understanding the implications of using jQuery in a web application.
I mentioned above that the case of using jQuery as a library requirement is left out of this question so it's not too broad. I also ask to leave out comparing jQuery with other similar libraries.
In order for responses to qualify as objective and non-opinion based, they must include substantiated methodological/technical arguments focused on the implication of using or not using jQuery.
Translation of the introduction of You may not need jQuery
jQuery and its cousins are great and should of course be used if they make it easier to develop your application.
However, if you are developing a library, please take a moment to consider whether you really need jQuery as a dependency. Maybe you can include a few lines of useful code and waive the requirement. If you're just targeting your library to newer browsers, you won't need anything more than what the browser already comes with.
At the very least, make sure you know what jQuery is doing for you , and what it isn't. Some developers think jQuery protects us from a big browser incompatibility demon when, in truth, post-IE8 browsers are pretty easy to handle on their own.