JediCP
It doesn't help that both departments use the same headed paper!
The simple way to think about it is the DWP are the people who do all of the management of your claim (e.g. new claims and renewals), their offices do not need to be near you, as they each cover a large regional area and Job Centre Plus (JC+) are the people who deal with the all the activities you need to do as part of your claim (e.g. WFIs WRA and Job Seeking), their offices will be local to you or in a nearby town.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems 
