Web Services should be tested before testing the functionality via the user interface.
As per one of the articles that I read about web service testing; 70% of testing related to any functionality can be considered as Done if web services related to that functionality have been thoroughly tested.
Reasons for doing the web service testing prior to functional testing (there may be many though):
Many times, there are different teams working on the development of UI and web services.
The UI of the application under development is not finalized.
There may be other reasons as well but these two are on top of my head right now.
In above two scenarios, it makes a lot of sense to do the web services testing separately without waiting for the UI. So, that most of the issues related to web services are caught even before the UI is ready and integrated.
This saves a lot of time when the integrated application is tested from UI. This should ideally result is lesser defects during testing which further results in lesser rework.
And even if the UI is not finalized and is being changed continuously; you are still confident that there won't be any surprises at the time of testing because web services have been thoroughly tested.
However, in my opinion, it won’t be a huge value addition to test the web services if the UI is present. You might test a few things that a user is not allowed to do from UI but that doesn’t make a huge difference.