1. Write SharePoint functionality in PHP
Yes that's right, SharePoint 2013 functionality can now be written in PHP, or indeed any web standard language. Specifically the new SharePoint (hosted) apps model, in certain circumstances, allows the use of a whole host of languages. You won't be coding masterpages in PHP any time soon, but it is great to see Microsoft adopt the wider web development world in the way they have.
2. The Geolocation list field
It is possible (by writing code, not via the interface) to add a new type of field to a list - the geolocation field. This makes it really easy to add location data to a list, and then view that data on a Bing map within SharePoint. There are all kinds of useful applications for this (office locations for one) and it really helps make SharePoint 2013 feel like a modern application.
3. Share content easily using guest links
Anything in SharePoint Online can be shared with the outside world by using "Guest links". Anyone user can be invited to view or collaborate on specific sites, documents, or folders by clicking the relevant ‘share’ control.
4. SharePoint Online will soon be updated every 90 days
Microsoft recently updated the world on its plans for it's enterprise social tool Yammer and Office 365. Buried in that announcement was the news that by 2014 Microsoft plans to update Office 365 and SharePoint with new features "roughly every 90 days". This is great news for SharePoint users, and finally brings SharePoint into the modern world of (almost) constantly updated web apps.
5. FAST Search is now built in
Do you remember SharePoint FAST search? It was an add-on to SharePoint 2010 and brought with it lots of new features, including advanced search algorithms and content previews. Going back even earlier FAST was a totally separate Norwegian company, that wrote the original product. Microsoft bought them in 2008 and folded them and their products into SharePoint.
Well FAST search is now a built-in component of SharePoint 2013, no extra installs or licenses needed. The most obvious use of this new technology can be seen in the document library previews, but the standard search engine takes advantage of everything FAST has to offer as well.