Why would you use one over the other, for exposing an API for your Django app?
As the author of django-rest-framework, I've got an obvious bias ;) but my hopefully-fairly-objective opinion on this is something like:
In any case, both are good. I would probably characterise Tastypie as giving you a sensible set of defaults out of the box, and REST framework as being very nicely decoupled and flexible. If you're planning on investing a lot of time in the API, I'd def recommend browsing through the docs & codebase of each and trying to get a feel for which suits you more.
Obviously, there's also the 'Why TastyPie?' section in it's README, and the 'REST framework 3'.
See also Daniel Greenfeld's blog post on Choosing an API framework for Django, from May 2012 (Worth noting that this was still a few months before the big REST framework 2.0 release).
Also a couple of threads on Reddit with folks asking this same question, from Dec 2013 and July 2013.