It is hard to compete with such illustrious neighbors as Siena or San Gimignano, so Volterra probably deserves its second-tier billing when it comes to Tuscan destinations. The historic town is an impressive enough specimen of medieval architecture without offering a "killer" must-see feature.
What Volterra compensates with is its Etruscan and Roman history. The collection of Etruscan artifacts at the Mario Guarnacci museum is among the best in the world of its kind. You can visit an Etruscan Acropolis at the highest point of the Parco Archeologico, right next to a Roman Cistern. From the northern edge of the old town you can have an excellent view over the remains of the Roman Theater (and for most people that is more than enough of a look).
The medieval town is compact and centered on Piazza dei Priori, dominated by the palazzo of the same name. The palace holds only a minor interest for visiting, but the tower views are worth the effort. The cathedral of Santa Maria Assunta has a number of features common to basilicas in Tuscany, but if you have seen Santa Maria del Fiore in Florence or Santa Maria Assunta in Siena, you will likely rate Duomo di Volterra as "just ok". Beyond those obvious points of interest, there are a couple of lesser museums and palaces that I myself have never been to in half a dozen visits to Volterra over the years.
Its lower profile aside, Volterra can still be pretty busy with tourists in season. People tend to gravitate towards aforementioned Piazza dei Priori or Via Antonio Gramschi, where the biggest concentration of restaurants is. That makes Parco Archeologico all the more delightful. A rare large green space smack in the middle of a medieval city, it feels relatively serene even in high season. For those with kids in tow, the playground in the park, rather average in the absolute sense, will offer an invaluable diversion.
Parking is always at a premium in popular Tuscan hill towns during the high season. In Volterra, it is undoubtedly to your advantage to park your car at the foot of the hill, and ascend to the city on foot. I most frequently use unpaved Parcheggio Docciola and walk up the stairs to Piazza XX Settembre through Porta e Fonti Docciola. This entry point deposits you just a few hundred meters from the central piazza, but you will be surprised by the contrast in the level of crowds or absence thereof.
I've seen towns that are much less worth of the recognition that are already inscribed on the WH list. Volterra would get three and a half stars from me.