Sabbatini and Sabatini have different pronunciations in Italian, since the double b requires a short labial stop between them. The same happens with Sabattini and Sabatini.
The name is a patronymic, i.e., it originated from the name of an ascendant person, and it is related to sabato, Italian for Saturday, indirectly from shabbat (rest), the weekly day of rest holiday for several religions, probably because the person was born in a Saturday. Due to the reference to the Jewish holiday, it has been speculated that people with this surname were New Christians or marranos (christianized Jews in the medieval times), but this is so far unsubstantiated. What is known for sure is that gentile families used to give the name Sabbato or Sabbatin to children who were born on Saturdays, as they used also Domenico to baptize children born on Sundays. Sabbatini is therefore the plural form used to name the descendants of someone who was named Sabbatin, Sabbatino or Sabbato.
Records indicate that the specific Sabbatini surname appeared for the first time in a noble family in Bologna, in the form Sabbadini. Since the 13th century it is a very specific surname of the region of Marche. Today, it is present also in the Abruzzo region (mainly as Sabatini, but also as Sabatino, Sabatini and Sabbatini) and in the Campania region, around Naples (where the more rare surname Sabbatino is present).
Due to the strong immigration of Italians in the late 19th century and early 20th century, the Sabbatini family name and its variations are strongly present in Brazil (mostly in the Southern and Southeastern regions, specially in the cities of Campinas and Valinhos), Uruguay, Argentina, Chile, Australia and the USA (with higher concentrations in the states of New York, Pennsylvania, Tennessee, Missouri and California).