tldr: The difference are significant but it isn't just a case of JSX vs Templates or large API vs small API. The choices that React and Vue devs have made may have consequences down the road that you may not realise when you first pick up one or the other.
React and Vue can accomplish much the same thing, they just go about it in very different ways. From some of the responses I wonder if people have actually used both libraries for anything beyond simple components as that is when you really start to see the difference outside of relatively superficial differences.
As you may or not know, Vue pushes the idea of Single File Components. A file with a separate block for your template, script and styles. Some people find this comforting as there is an element of familiarity with traditional front-end development. I personally really like them, not because it 'looks like html' but because it creates structural landmarks on the page which make it quicker to navigate the file itself. React's JS/JSX combo works fine but things are a little more muddled (especially when you add styled components to the mix). You get used to it regardless. It is worth pointing out that the use of SFCs requires a build step, while Vue itself does not. I doubt this is a concern if you are considering React as an option because JSX requires a build step. if you need the ability to drop some script files into a project without a build step then React is not really an option unless you're willing to drop JSX and use
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