Google broke YouTube for Firefox users right before the last Christmas holidays. Mozilla pushed out Firefox updates over the holidays that faked the UA as a workaround.
Google Inbox didn't work on Firefox initially, because Google claimed Firefox was too slow. When reported to Mozilla, it was fixed in a few hours, and they found that Chrome's implementation of the "thing that was too slow" was actually broken and not spec compliant.
Hangouts (used to?) require a plugin in Firefox, and just uses WebRTC in Chrome.
And so on...(sigh)
https://bugzilla.mozilla.org/show_bug.cgi?id=1087963
Here's the Firefox bug report about Google Docs being 3-9% faster, in a particular performance benchmark, when Firefox spoofs the Chrome UA:
Google Hangouts uses WebRTC and NaCl without plugins in Chrome, but requires an NPAPI plugin or ActiveX Control (confusingly called the "Google Talk Plugin") in other browsers. Support for any browser other than Chrome is not listed on the Hangouts home page. You must search their KB to find the plugin installer.
As someone who has never used Chrome, I'd love to understand what exactly "works better" means. I've never had problems with any of those sites using Firefox and Safari.