Several different manuscript versions of Pliny's letters have survived to the present day, and they have various different dates recorded for the month and day. This means that in the long succession of copies of copies of copies through the ages, some changes were introduced into the original text, for whatever reason, and now it is not clear which is "correct." The most common text today says Nonum kal. Septembres hora fere septima -- that is, "nine days before the kalends of September, at about the 7th hour of the day (~noon-2pm, counting from dawn)" aka August 24th. The Romans did not number each day of the month as we do, but counted backwards or forwards from set marker days for each month, the kalends at the beginning of the month and the ides from the middle of the month (thus the famous Ides of March). The fact that the Pliny manuscript versions differ might suggest that there was a disagreement about the date even many centuries ago. Perhaps some compiler in the 7th century CE had another source, now lost, which recorded a different month and day, and so he amended his copy of the Pliny letter to match, and thus created a secondary version. Who knows. Here is a serviceable translation of Pliny's letter.

The Italian team found an inscription (a charcoal scribble on the wall of a ruined building) which includes a date of 16 days before the kalends of November, which would be October 17th. Since it would be theoretically impossible to write that on a wall of a collapsed building buried dozens of meters below a still-smoking sea of volcanic debris, the excavators now are convinced the actual date of the eruption of October 24th (one of the competing dates). There is some other circumstantial evidence, as well, such as the carbonized remains of autumnal harvest vegetables which wouldn't (they think) have been already harvested and in people's houses by August 24th.