When talking with Salvatore, Brother William seems surprised to find that Salvatore eats the rats he catches. Rats were commonly eaten by the poor in the Middle Ages. Brother William would have been more surprised had someone catching rats not eaten them.
The secret message on the parchment is exposed three times. The translator heated it to reveal the location of the library, William of Baskerville heated it again when he was in the scriptorium and yet again to show the others the message. When a message is written in lemon juice, heating it will cause it to be exposed because the sugar in the juice is caramelized. It does not disappear again.
The abbot greets Brother Cuthbert of Winchester as "Your Grace". This form of address would only be correct for a high level noble (who would not be a member of an order of poor monks) or in England for a high member of the Catholic Church. It would not be appropriate for a monk in Italy.
At several points in the film, individuals enter another's cell (room) rather freely. In ancient religious rules, entering another monk's cell without permission was normally forbidden, even punishable by excommunication.
The goof item below may give away important plot points.
The first time in the library, Adso unravels his robe to help find the way out. Although Brother William and Adso are shown with only one robe each, Adso's robe is back to normal after exiting the library.