In Britain the document is a covering letter and it covers a CV, not a resume. The difference is more than vocabulary: the British letter is shorter than the American one, more formal in its frame and more restrained in its claims. Two hundred to three hundred and fifty words on a single A4 side is the norm, and a letter that runs to a second page is read as an inability to summarise.
One rule is enforced by anyone who went to school here: the sign-off must match the greeting. If you opened 'Dear Ms Alvarez' you close 'Yours sincerely'; if you opened 'Dear Sir or Madam' you close 'Yours faithfully'. Mixing them is the most commonly spotted error in a British application, and it is exactly the kind of small carelessness a covering letter is supposed to disprove.
Recruitment here is also more specification-driven than in the United States, particularly in the NHS, universities, local government and the civil service. Those employers publish a person specification with essential and desirable criteria, and the letter — sometimes retitled a supporting statement — is expected to walk through the essential ones with evidence. In the private sector the same discipline simply takes a shorter form: pick the two or three requirements that matter most and answer them.