All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
The order to differ from the Jews concerning fasting the day of ‘Aashooraa’ (10th of Muharram) such that one fasts another day along with it [whether before or after it] came during the last period of the life of the Prophet, , as the questioner mentioned.
Ibn Hajar from the Shaafi'i school of jurisprudence said in his book Fat-h Al-Baari, “There is another narration reported by Ahmad that was narrated by Ibn ‘Abbaas in which the Prophet said: ''Fast on the day of ‘Ashooraa’ and be different from the Jews, fast one day before it or after it.” This was towards the end of his life. The Prophet used to like to act in accordance with the People of the Book in matters where he was not ordered to do otherwise – especially when this would be contradicting the practice of the idolaters. However, when Makkah was conquered and Islam became widespread, the Prophet would differ from the People of the Book, as was narrated in a Saheeh Hadeeth (authentic narration). Therefore the Prophet did the same as the People of the Book did initially, and said: 'We are worthier of Moosa (Moses) than them.' Then he liked to differ from them and ordered us to add one day before ‘Aashooraa’ or after it in order to be different from them." [End of quote]
For more benefit, please refer to Fatwa 45817.
Allaah Knows best.