I'm actively using both frameworks on separate projects and have roughly 1 year of experience with Tastypie and 2 months of experience with DRF. Both have strengths and weaknesses with very similar feature sets, although I personally prefer DRF, backed up by my choice to use it over Tastypie when starting a new project 2 months ago.
Tastypie has a rather monolithic structure (most of the logic is wrapped in a single resource object) that passes responsibility from function to function. I find the code hard to read and mentally compartmentalize the various components. It does a great job providing basic functionality out of the box, but as soon as you start to do anything complex you will find yourself repeatedly digging through Tastypie source code to track down behavior. Tastypie approach to URLs is also rather limiting.
DRF uses Django class based views and generally follows Django's design patterns. I find the code much easier to understand and more inline with the direction Django is heading. The module class structure makes it much easier to build reusable components. One downside with DRF is their permissions system, it has a rather large oversight for multiple object permissions. DRF completely excludes the queryset from the permission decision which leads to some terrible workarounds (I'm not sure how you can make an informed permission decision on a set of objects without access to the set of objects themselves).
The documentation and community on both is rather strong, although there is more information on Tastypie as they've been around longer. As a point of reference, Tastypie has ~750 questions on stackoverflow vs 153 for django-rest-framework. Both are actively maintained, but Tastypie's design architecture handcuffs it's ability to move forward.
tl;dr; My $0.02 - Tastypie was the standard, DRF is becoming the standard. I would personally choose DRF over Tastypie for any future project.