Well, it sets the 'tone' of the site. Just like the conversation will be different with the same group of people depending on whether they're all dressed in tshirts or if they're all dressed in formalwear... in each case, they're the same people 'underneath', but the image each sees of others and each's self-image will differ.
There's an additional layer of mapping added when using shortnames - the user has to remember a whole list of "@donkey2001,Joe Smith" tuples; most people (and especially non-geeks) like and operate better with fewer layers of abstraction.
The usage of a full name on Facebook makes explicit the roughly one-to-one correspondence between accounts and actual-people that is expected by users and required by the service; Twitter users openly create multiple profiles with different names and even identities, and the service and tools encourage that. In that sense, Twitter has more of a sense of a masquerade or Halloween party and less like a business meeting or high-school reunion.
Each has its strengths and its place, and the longer I use both, the less I feel like there's direct competition between the two.
