React is, from my perspective, an attempt to abstract away a lot of DOM, with the objective of decreasing the size of the solution space to common problems.
The idea is that two engineers working in two different parts of a company would write more or less the same code in react, whereas the jquery solutions would have been very divergent. React enforces patterns and makes it harder to write bad code. That’s the dream, at least!
I wouldn’t compare the two directly, they’re solving totally different problems. Jquery makes writing vanilla JS less verbose. React is more like an engine that’s written in JS and programmable with JS, with the goal of making browser code more composable and opinionated (wrt data flow).
Long answer is that they don't do the same thing. I think jQuery was an incredibly clever and durable library and became so ubiquitous that most of it's most valuable features got folded into the JavaScript and CSS standards and no longer require dependency on a library. React is a modern way of structuring components. It's a fresh learning curve, but it's actually pretty intuitive and very useful.
react is a compiles-to-plain-js framework for building data heavy complex applications in the web browser.
React has its own template language built into your functions, quite opinionated data flow paradigms, and tries to abstract all of the direct attacking the DOM away into passing pure data around.
JQuery gives you better tools for hitting the DOM, but still is just a set of tools, not a development methodology.
This is simplified, obviously, but trying. :-)
In 2010 terms: let's say you want to write a web app. jQuery : your library of useful PHP functions :: React : Ruby on Rails. That analogy breaks down in many obvious ways (React isn't MVC and [modulo JSX transpilation and whatnot] uses the same language as jQuery, Rails is not easy to reason about especially if you're in 2010 and method_missing isn't considered harmful yet), but hopefully the basic idea comes through - library of functions vs. very opinionated framework that facilitates rapid development.
jQuery: Add some javascript features like ajax/animation/some DOM manipulation to a (dynamic/static) HTML page.
React: Create the dynamic HTML from scratch, that will still do ajax/animation etc.
Server-side React is a great way to generate HTML.