I think puppeteer is an interesting project, but right now it's Chrome-only and therefore pretty much useless for cross-browser testing. Even with support for Firefox, it would still lack support for Safari Desktop, Safari Mobile, Internet Explorer and Microsoft Edge. It also won't allow you to run the same tests against real devices.

I think Puppeteer is likely the superior choice for any browser-automation task apart from cross-browser testing. But if you want to make sure that your website is working for your users, using a Framework that uses the standardized W3C Webdriver API is the far better choice, unless you only want to support Chrome.