The trip I was looking at (Thomas Cook so probably could find cheaper) included 7 nights in a hotel in central Rome, entrance to the Vatican museum, Sisteen Chapel, Collesium(sp?) etc and another trip to Pompeii as well as a night at the hard rock cafe - that came to around £1800.
NY is more pricey but I suppose its understandable really.
My two all time favourite cities, but unless your DCs are fashionistas or very into the ancient world or fine art (rare but possible in teens) I'm sure they'd prefer New York.
But if NY would be too much of a financial squeeze I'm sure you would all have a lovely time in Rome as well - you'd have to be a real brat to turn your nose up at a trip to Italy, and I'm sure your DC aren't.
ROME... they can do New York when they are big enough to order drinks in a bar, and have stopped growing so they can buy tonnes of clothes. in the meantime Rome is great for buzz and pizza and pasta and beautiful people..