IMO, AsyncTask was written to provide a convenient, easy-to-use way to achieve background processing in Android apps, without worrying too much about the low-level details(threads, message loops etc). It provides callback methods that help to schedule tasks and also to easily update the UI whenever required.
However, it is important to note that when using AsyncTask, a developer is submitting to its limitations, which resulted because of the design decisions that the author of the class took. For e.g. I recently found out that there is a limit to the number of jobs that can be scheduled using AsyncTasks.
Handler is more transparent of the two and probably gives you more freedom; so if you want more control on things you would choose Handler otherwise AsynTask will work just fine.