Common usage in the UK is that a postman of the Royal Mail Service delivers the post, and someone may post a letter (see BrE Ngram), whereas in the USA, usage has become equally common that a mailman of the United States Post Office delivers the mail (AmE Ngram). Why do both forms of English have the same two words and have parallel but reversed inconsistencies between the name of the Service and the objects manipulated by that Service? And what do Canadians do? (There's no "Canadian English" option in Ngram!)
I feel the consensus opinion is that
(1) it's possibly/probably true that "mail" is used more - in general - in the USA than in Britain. I really feel that's about all you can say about usage in bre/ame in this case.
(2) the specific, clean 'reversal' you point out (mail/post on one side, post/mail on the other) is probably spurious; it does not exist.
(3) it's very unlikely there would be a clean explanation of any difference in usage of the words; fwiw nobody at hand has one or could find one :O
I fear that's it!