@gr9g What relation to reality, if any, these things have ever had as depicted in the manga or anime we've all come across, I can't say. But I'd assume the separation of the public morals committee from the student council has a couple of possible reasons, including things like having them in a position to keep each other in check. If for some reason the student council is goofing off, the public morals guys could speak up about it or vice versa.
Another thing that comes to mind as a possibility comes from the naming for the public morals committee. In some of the manga I've read, there are various committees with more specific functions than the more "generalist" function of the student council. Some of these have been shown as basically subordinate to the student council, so it could just be delegation. While the president and then are dealing with paperwork and organizing things, the public morals committee would be like the foot soldiers or the beat cops, out on the "front line."
Those two ideas are pretty much mutually exclusive but I feel like I've seen both depictions.