The Bankers Special was built on the short Police Positive frame. It could be purchased in either .38 Colt New Police (same as .38 S&W) or .22 LR.
The Detective Special was made using the longer Police Positive Special frame, and normally came as a .38 Special, although following World War Two, .32 Colt New Police (same as .32 S&W Long) was added to the list.
The only real difference was the length of the frame and cylinder. The Bankers Special was discontinued following World War Two.