I've been to Italy, Istanbul, and India.
India would certainly be a different experience, and would be fine if you're looking to see how much of the poorer world lives. I've been there three times and would never, ever think about going there on a vacation. Driving around the slums, seeing poverty, suffering, etc., isn't an enjoyable week for me. YMMV.
If it's just Rome vs. Istanbul, it's a bit of a toss-up. Both are fantastic cities. If it's more than a few days and you have the ability to travel around a bit, I'd strongly recommend Italy, heading down the Amalfi coast and over to Capri for a day. Though you can't go wrong with Venice, Florence, or numerous others.
Personally, I'd skip Vietnam, as I suspect it will be like India, but that's solely based on reputation and external observation.