At the beginning, there were quite a few issues and it was only Chrome that properly supported the webp usage, so there wasn’t much point.
Recently however, webp is now a lot more supported in most of the major browsers (only people still using IE or old versions of safari generally have issues).
I would recommend reading through this article that helps with setting up fallbacks etc. such as using the picture element and having a jpeg fallback option if the browser doesn’t support it.
This does mean if you want to have a fallback for older browsers, you will need to store both the webp and the jpeg version.
For sites that use a lot of images, it definitely helps with the page load speed/SEO and something I would look into.
Cheers
Tim