A brief search shows that an original sense of the word "data bank", was to refer to an organisation that operated a data-processing and storage facility, by analogy to the high street bank that operates a money-processing and storage facility.
I would speculate that this was quickly slurred to refer to internal functions which provided the same facility to a single organisation (an organisation which was not itself a "data bank"), and also becoming associated with the computer hardware itself.
The term "data bank" nowadays is obsolete, except as @amon mentioned, where it is used in the sense of referring to an organisation whose main purpose is maintaining a data archive of perceived civic or scientific importance - similar to a library, but perhaps emphasising that what is stored is primarily structured data rather than books, articles, and other narrative in natural language.
A "database" by contrast does not mean an organisation, a function, or equipment. It is a volume of fully-structured data typically stored on a computer, and perhaps conflated with the software which manages that volume.
The reason "data bank" has fallen into disuse is because nowadays it is just synonymous with being in business. Everyone has a computer and an IT function (whose roles are often through of primarily as engineering and maintenance of hardware and software, not just curation of data).