What is the historical difference between Quba Mosque, which was founded on piety, and the current Noble Prophetic Sanctuary, and why is there this confusion between them in historical narratives?

There is no confusion in the historical accounts between Quba Mosque and the Prophet's Mosque. Quba Mosque is located outside the city of Madinah, about two miles from it. The Prophet, peace be upon him, stayed in Quba upon his arrival in Madinah, remaining there for four days during which he built Quba Mosque and prayed in it. He then proceeded to Madinah and built his mosque where his she-camel knelt.

No scholar has ever claimed that Quba Mosque is the Prophet's Mosque, and the distinction between them is clear in the prophetic hadiths.

Scholars have differed regarding the meaning of the verse (لَمَسْجِدٌ أُسِّسَ عَلَى التَّقْوَى مِنْ أَوَّلِ يَوْمٍ أَحَقُّ أَنْ تَقُومَ فِيهِ) (Al-Tawbah: 108). Some said it refers to Quba Mosque, others said it refers to the Prophet's Mosque, and some said the verse includes both. This latter view is what Ibn Hajar and others chose. Shaykh al-Islam Ibn Taymiyyah affirmed that Quba Mosque was founded on piety, and the Prophet's Mosque was also founded on it, being more complete in that regard, so the verse encompasses both.