Is a "Bank Holiday" treated any differently to a "weekend day" by the banks?

Armorica

Legally bank and public holidays are different. Bank holidays are created first and foremost by statute (and for the purpose of pausing financial dealings), or Royal Proclamation. Public holidays were created first and foremost by common practice and history. And even the same "day" can be a bank holiday one year, and a public holiday another depending on the day of the week...

But the practical impact is basically the same in terms of what is open and employee rights (well, ish - some employee holiday is a statutory requirement, some flows from employee contracts).  Less so about whether someone works for a bank - the act is about dealings with customers (especially in the old days about certain financial instruments).

A clear guide is https://researchbriefings.files.parliament.uk/documents/SN06170/SN06170.pdf