As of today (nov. 2013) Dynatree is the stable version.
While the core functionality in Fancytree is already pretty stable too, there still may be some changes to the API and markup. I am planning to release Fancytree early next year.
So if you want to be on the save side, use Dynatree. If you are fine with core functionality, want to help to test and improve this open source project and are ready to modify your code every now and then: use Fancytree. (Disclaimer: I am the author of both.)
Update 2014-05-01
Today I released v2.0.0