The name pretty much sums up the difference: asynchronous programming vs. parallel programming. But in F# you can mix and match.
F# Asynchronous Workflows
F# async workflows are helpful when you want to have code execute asynchronously, that is starting a task and not waiting around for the final result. The most common usage of this is IO operations. Having your thread sit there in an idle loop waiting for your hard disk to finish writing wastes resources.
If you began the write operation asynchronously you can suspend the thread and have it woken up later by a hardware interrupt.
Task Parallel Library
The Task Parallel Library in .NET 4.0 abstracts the notion of a task - such as decoding an MP3, or reading some results from a database. In these situations you actually want the result of the computation and at some point in time later are waiting for the operation's result. (By accessing the .Result property.)
You can easily mix and match these concepts. Such as doing all of your IO operations in a TPL Task object. To the programmer you have abstracted the need to 'deal with' that extra thread, but under the covers you're wasting resources.
Like wise you can create a series of F# async workflows and run them in parallel (Async.Parallel) but then you need to wait for the final result (Async.RunSynchronously). This frees you from needing to explicitly start all the tasks, but really you are just performing the computation in parallel.
In my experience I find that the TPL is more useful because usually I want to execute N operations in parallel. However, F# async workflows are ideal when there is something that is going on 'behind the scenes' such as a Reactive Agent or Mailbox type thing. (You send something a message, it processes it and sends it back.)
Hope that helps.