While you're here at the Vatican, you will, of course, see the major attractions. Some lesser-known "sights" are located here also. One is the post office. Italian mail service is notoriously slow, so the efficient post office offered here is the place to send off your postcards if you would like them arrive home before your next trip to Rome. (This is not much of an exaggeration. My sister sent me a postcard from Italy and it took 6 weeks after her return home to reach me!) I know people in Rome who will take the Metro to the Vatican solely to utilize the postal service here instead of using the one in their neighborhood.
Another is the bathrooms. Okay, I know you're not going to head over to the Vatican solely for the bathrooms, but in a city where public restrooms are scarce, you may as well enjoy it while you're here! Nice, clean, large, free....enough said!
And, then there is my favorite part of coming here. Watching the people. I like to stand in the wide piazza and observe the people - a multitude of nationalities, backgrounds, faiths, languages, making pilgrimages for a multitude of reasons. It always brings to mind the verse in Revelation "you purchased men for God from every tribe and language and people and nation."