I'd think a housekeeper is simply doing more hours and on a set salary. A housekeeper would be expected to organise the house; think tidying, cleaning, cooking, laundry, ironing, putting away clothes, running errands, shopping, organising household maintenance and repairs.
I've worked as a cleaner and I don't mind tidying up. However, I charge by the hour and time slots are agreed beforehand, so if I spend an hour putting away toys or washing up, I'm going to either spend less time cleaning or charge an extra hour.
I also don't know how people want their house organised. I don't always know where everything should be tidied away to, which kitchen cupboards are used for what or it's sometimes confusing who's clothes are who's.
Most cleaners will accommodate some housekeeping tasks though. I've incorporated ironing, laundry, bed making into my role; charged at the same hourly rate but obviously it's going to take far longer.