I've been a nanny/babysitter for 20 years now. I think the main difference is frequency of care-taking. If you have someone who watches your kids every now and then or maybe even weekly, but for a couple of hours, it's babysitting. If it's a regular job (20-40 hours week or something) it's nannying. That's my understanding. I don't know what others think or if it's even defined somewhere.
I may be able to help you out as well, if you need. I'm a SAHM to a 2yo boy. If you wouldn't mind them having 'sleepovers' here (I have many beds and pack-n-plays), I could probably convince my husband.