If you're comfortable with long walks or public transit, you could hit a bunch of sites in Rome on your own. If you're doing that, you will need to get transportation to Rome as well, either by taking the train for Civitavecchia or doing the Rome on Your Own excursion.
My family (with me, husband, and kids 10 and 14) did a private day tour with Rome in Limo that included the Coliseum, Pantheon, Trevi Fountain, Spanish Steps, a private tour of the Vatican (including the Sistine Chapel and St. Peter's), a pizza lunch, and 1 or 2 other places for about the same as we would have spent for an excursion through
Disney Cruise. It was a long day, but we saw a lot.
An advantage to booking through
DCL is that you will make it back to the ship on time or DCL will get you to the next port. However, our driver was in constant touch with his office to be aware of any traffic issues, as was the driver from Joe Banana Tours who we used in Naples on a different cruise. If you use a reputable company, getting back is unlikely to be an issue in Europe. Major traffic tie-ups will affect everyone, including Disney buses.
A disadvantage to booking through DCL for a tour with multiple stops is that you will be waiting until everyone is back before you can move on, so you will probably see less than you would on your own.