Nothing much. OKHttp async is OKHttp API driven. So as long as you bundle the jars together for all platforms you should be good. AsyncTask is Android way of doing things.
However since Honeycomb Async task runs the tasks sequentially and not in parallel. This means that though the execute method of AsyncTask spans a new thread which runs your job away from the UI thread but all the tasks sent to one AsyncTask run in the same spanned thread.
So for 3 tasks submitted u don't get 3 threads they all run sequentially on a single spanned thread. With OKHttp you can achieve true parallelism using callbacks and async GET and POST.
Though you can do true parallelism in AsyncTask methods as well (check the overloaded execute methods in AsyncTask) but default Android behavior is not to do so.