As for concrete examples, Hangouts only works in non-Chrome browsers (including ones with WebRTC support) if you install a Google-provided binary blob. Which you may not be able to do.
Gmail only supports offline access in Chrome (see https://support.google.com/mail/answer/6557?hl=en the "two exceptions" bit). Whether not having offline access to your mail counts as mail "not working" is up to you, I guess; for me it counts as "not working".
Various Google properties use UA sniffing to deliver degraded content to non-Chrome browsers. https://bugzilla.mozilla.org/show_bug.cgi?id=921532#c9 is an example.
https://bugzilla.mozilla.org/show_bug.cgi?id=973754 is an example where as far as I can tell they built the feature around non-standard Chrome-only functionality even though Firefox supports the standard version.
Google news menus don't work in standards-compliant browsers because they rely on a Chrome/WebKit bug. See https://bugzilla.mozilla.org/show_bug.cgi?id=1083932
Google patent search uses UA sniffing and locks out various browsers as a result. See https://bugzilla.mozilla.org/show_bug.cgi?id=1013702
Google Translate will fail to work in Firefox unless you have Flash installed (good luck on Mobile).... or spoof the Chrome UA string. See https://bugzilla.mozilla.org/show_bug.cgi?id=976013
They do fix these bugs sometimes (the UA sniffing ones, where they just got the sniffing flat out wrong, tend to get fixed once someone diagnoses them). And sometimes not.