A nanny is hired to help with chores related to the children in the house, meaning the work is not limited to just watching the children. The babysitter's main responsibility is to take care of the children, often for a set amount of hours only.
A babysitter is often hired by the hour. A nanny usually lives in the house with a family and works only for one family. Normally, a babysitter works for many different families. A nanny is supposed to help children with their homework, fix their food and perform other household tasks that are related to children.