Lazard has given the correct advice. Always exit/enter a given country on the same passport. That just makes sense if you think about it. If you enter on one and exit on another than there is no indication that the holder of the entry passport ever left the country. In other words, it could lead to a belief you had overstayed.
US citizens are required by law to exit and enter the US on their US passport. Not a problem since the benefit of the second passport never has anything to do with entering/exiting the US anyway.
Entering Italy on an Italian passport has the benefit of the shorter line at Customs & Immigration. Just make sure to follow the enter/exit on the same passport rule.
Entering a third country (not US or Italy/EU) the benefit as stated may be related to a visa. You may not require one with one passport while entering on the other passport would require one. You may pay a lower visa fee with one vs. another. For example, an Italian passport holder is not required to have a visa to enter Turkey but a US passport holder is.
The real biggy though in having got her Italian passport is that she now has the right to live and work in any EU member country. She is also exempt from the Schengen 90 days in 180 rule that a US citizen has to comply with. So if she wants to spend an extended amount of time in Europe she can do so.