The new moon is seen a full day, sometimes 2 days after it says it should be on the Jewish calendar. Why does no one care?
In the days that we had Sanhedrin, the large assembly of seventy judges and the Kohen Gadol, Rosh Chodesh was decided by Sanhedrin. Each new month was arranged when witnesses saw the new moon and testified to the Sanhedrin regarding the sighting. Sanhedrin would then decide when to sanctify the new month and the news was passed from town to town. However, it must be noted that Sanhedrin had the authority to decide if the new month should begin on the day that the witnesses testified or the following day. They could push Rosh Chodesh off or pull it back, and even create leap years when necessary.
During the period of the Amoraim, Hillel anticipated that the Sanhedrin would soon cease to function, effectively disrupting the Jewish calendar and all subsequent holidays. Hillel, therefore, created a new calendar with a nineteen-year cycle and many complexities that would remain extremely accurate over the next two thousand years.
Because of the need for a set calendar, the actual date of Rosh Chodesh does not usually coincide with the new moon. This is not an issue though, since the
Rosh Chodesh was always decided by the sages and did not have to be on the actual date of the new moon.
I hope this clears the topic up a little.
All the best
Rabbi Shay Tahan