I know that the assignment of chapters and verses to the translated text of the bible is a Medieval innovation. My understanding is that it was invented by non-Jewish scholars. However, in several place in the Plaut commentary (I can only read the English), R. Plaut says that the Jewish division of verses differs slightly from the Christian version. Why is that?
The Christian division of Chapters and Verses follows their understanding of the Torah text which sometimes conflicts with the Jewish understanding.
There was a time when Biblical chapters and verses did not exist. There was no way of accurately pinpointing a verse except by indicating whether it was from Bereishis or Shemos, or perhaps mentioning that it was at the beginning or end of a weekly Sidra. To locate pesukim in olden times, you had to have a basic knowledge of the entire Tanach. Even though Hashem gave us the Torah divided into parshiyos (identifiable by gaps at the beginning or middle of lines) and pesukim, no one ever ascribed numbers to them, and the Sedarim later used in the three year Krias HaTorah cycle and the fifty-four weekly sidros used in our time were identified by name and never by number. Furthermore, no one ever created a chapter system based on the seven aliyos of the weekly parshiyos.
A Church Invention
Credit for dividing the Tanach into convenient chapter and verse generally goes to Cardinal Stephen Langton of France and England, who served as the Archbishop of Canterbury. He is thought to have created the chapters and verses in about 1205, which are used until this day. This was not too difficult a job since 617 out of his 779 chapters coincide with the parshiyos that existed since Sinai, while of the 162 chapters that he invented, many are illogical, while some border on the heretical.