It's great that you can add web parts to the public site – but in one of those pointless moves seemingly designed to needlessly frustrate users – MSFT decided to limit the web parts made available. So here's how to overcome that restriction. I've used this technique to add a lot of different web parts to the public site – in this example I'm going to use the "Picture Library Slideshow Web Part" which was suggested by markgi_MSFT in the 365 forum.
Export the web part.
First you have to locate the web part. It's probably going to be in your team site web part library:
Click the "cog" in the top right corner and navigate to Site settings»Web designer galleries»Web parts.
Locate MSPictureLibrarySlideshow.webpart and click it to download.
TIP: If you can't find it, you can download it here: https://wb2-web.sharepoint.com/Docs/MSPictureLibrarySlideshow.webpart
Import the web part.
In another brilliant move, MSFT hid the public site web part gallery, so you'll have to enter this url to open it: yourpublicsitedomain.com/_catalogs/wp/Forms/AllItems.aspx. Import the web part by navigating to where you saved it.
Now it should be available to add to a public page. Here's an example:
http://365new.webbrewers.com/Pages/Slideshow-web-part.aspx Here's how it should look:
Download Web Parts Here!
There are 72 web parts in a 2010 Enterprise account. If you can't find a web part in your Small Business team site, open up a trial Enterprise account and you'll find all 72!. You can drag and drop all of them over to your public site if you open the public and team site web part galleries in either windows explorer or Sharepoint Designer. Or you can download a couple of popular ones here: