The Nanny

A baby sitter is a part-time position. They are typically girls in the 14-23 year old range that do it while in school. A Nanny will usually live with the family, or be resident through most the day and leave only at night. Think of a nanny as an au pair.

http://www.answers.com/topic/au-pair-1&method=6

A nanny is a childcare professional whose is qualified and paid to look after one or more children from the same family, in their own home, usually on a fulltime basis.

A babysitter is less likely to be professional, and is required less frequently and for shorter periods of time, e.g. an evening. The term is quite flexible, covering a whole range of different types of carer, from a neighbour or family member who babysits for little or no money, to someone making a living from offering babysitting as a service.

Thank you J. and Dave for taking the time.

In the US, it doesn't really matter. It is used alternately, although rich people prefer to use "nanny" than "babysitter".

Thank you Maevi, for your insight.

What part of the US do you live in?

I live in the US, and I've rarely heard people refer to even "glorified" babysitters as nannies. "Nanny" is very close in meaning to "au pair" or "governess," who is the professional caretaker of the kids on a full-time basis, sort of in loco parentis, as Dave says.

On the very few occasions when I've heard "nanny" used for "babysitter," it was for a temporary gal to sit at home with the kids and take care of the house while Mom and Dad were on vacation for a week or two.

As a recent college graduate in the US seeking a summer "nanny" position, this is my read on the subject:
A nanny is someone with hopes of full-time (or even major part-time) REGULAR work in accordance to a pseudo-contract. It usually mean 20+ hours a week, and indeed, is a "glorified babysitter" in many senses.
I think it allows both the employer and the nanny to believe the work is more sophisticated, "resume-worthy." Baby-sitting is much less contractual.