Nanny: cares for children, teaches, possibly cooks and takes care of child-related housekeeping (make and clean up after lunch)
Housekeeper: cleans the home, possibly makes dinners ahead, possibly does family laundry, possibly errands
House manager: schedules and supervises services, takes care of household ordering, takes care of pet care and vet trips, takes care of plants, may research and schedule vacations and household travel, supervises housekeeping, supervises childcare, liaises with personal assistant/estate manager, runs errands, etc. May arrange logistics for moving, parties, etc.