I think that would be the Via Venetto neighborhood which has the exclusive shops and embassies.

Piazza Navona/Pantheon is a great neighborhood to stroll. Definitely visit Navona at night.

I stayed on the Aventine Hill at Villa San Pio. Rome is busy and this was a quite residential neighborhood. We had a huge room with a private balcony with a free breakfast brought to our room. The hotel has received many favorable reviews on Fodors. I think they are on Charming Italy also. They have some sister properties that are less expensive - I think these dont have air conditioning.