Every cool new thing targetting the browser uses jQuery. How this relates to using a big framework like React?
Take for example AngularJS. AngularUI just built a set of Boostrap directives so its not dependent on jQuery:
http://angular-ui.github.io/bootstrap/
This guy says, "You might not need jQuery" and gives some good examples:
http://youmightnotneedjquery.com/
I have a co-worker that will go to his grave thinking VanillaJS is much much better than jQuery. .
To try and answer your question, most jQuery apps have the business logic, the model and the UI all together. React (as I understand it) abstracts each of these out separately. This makes debugging, and maintaining a large scale app a lot easier. Also, jQuery is all about manipulating the DOM. React uses other methods to stay away from directly interacting with the DOM.
Angular has a version of jQuery embedded, and replaces that with the full version if it is included in the page. AngularUI was not there to reduce dependencies on jQuery (UI, which is different than jQuery Core), but rather to provide the UI patterns in Angular directives so that developers didn't need to write their own over and over. That said, if you are new to Angular, my recommendation is to try not to use jQuery as that will force you to consider the "Angular" way of doing things rather than the "jQuery" way.
The guy who wrote You Might Not Need jQuery has also said that he was looking more at framework developers who don't need to make jQuery a dependency, not the average joe building a website. RWaldron, a jQuery contributor, put together a list of quite a few of the bugs that jQuery fixes for browser compatibility.
Don't get me wrong, Vanilla JS is great, and I love the site. But there are quite valid reasons for using jQuery and many of them are just as valid now as they were in the past. You can write "good" jQuery, it doesn't have to be "js soup".