Nowadays, none whatsoever. Historically, the British passport's 'Issuing Authority' was the entity handing over the passport on behalf of the monarch and their government. For example, the Post Office or the Home Office (the equivalent of their State Department). However, eventually, the bureaucrats realized that this made little sense since it implied that these "authorities" were somehow coequal to–gasp–the monarch themself, and they stopped the practice after that.