Web Safe Fonts
Google Fonts are served as web fonts as part of every Google Sites page.
Google Sites also allows site editors to select additional fonts, also available in Google Docs and Slides, even if not listed in Google Fonts. These fonts are licensed to Google for use on Google’s domains, but do not work with custom URLs.
However, a small selection of “web safe” fonts will work for site viewers using Microsoft and Apple operating systems. Site viewers using ChromeOS, Android, and other operating systems that do not have these “web safe” fonts installed will see alternative fallback fonts on sites with custom URLs.
Google Groups embeds
Google Groups embedded in Google Sites that have been published to a custom domain will not render properly.
Google Account authentication
Other features that depend on Google Account authentication may not work since a custom domain mapped site is no longer hosted on sites.google.com and cookies for *.google.com are not sent.
Sharing settings
Your site must be shared with the world for published viewers, or it will redirect to the sites.google.com URL for your site.