There is no verse in the Quran or no hadith in saheeh hadith books about the difference of men’s and women’s prayer. However, there is a hadith in the book named Marasil by Abu Dawood on this subject. Marasil is NOT the book of Abu Dawood which is among the Kutub al-Sitta.
Books of Marāsīl are collections of mursal ahadith. A mursal hadith is a narration in which a narrater omits his teacher and directly narrates from the Rasulallah (pbuh). Mursal hadith cannot be used as evidence to make judgment on a subject because the narration chain is discontinuous.
Current differences arise from the opinions of jurisprudence scholars. They assert that these differences enhance the covering (tasattur) of women. If you check the book Al-Hidayah by Burhan al-Din al-Marghinani, which is one of the main resources of Hanafi jurisprudence, you will find out that the main motive of the difference is claiming to be “more suitable to the tasattur of women”.
In conclusion, there is no verse or saheeh hadith to prove that there should be a difference between the prayer of men and women, or to prove that these differences are more suitable for the covering of women.