Shia Muslims follow the authentic narrations from Ahlul Bayt (AS) in every matter, while Sunni Muslims follow narrations from other than Ahlul Bayt (AS) like Abu Huraira, Ayisha, etc.
The most authentic narrations from the Progeny of the Prophet (SAWA) state his birth date as 17th of Rabi'ul Awaal.
'Sunni books have different birth dates which shows that Sunnis themselves don't have one date although many of them take 12th Rabi'ul Awwal as the birth day and death date of the Prophet (SAWA).
Obviously, no one knows the birth date and death day more than the family members,that us why we, Shia Muslims follow Ahlul Bayt (AS) in this matter and every matter.
'Wassalam.