Why were the Prophet's hadiths not collected and written down during his lifetime, and how can one distinguish between the authentic and the weak among them?

The Prophet, peace and blessings be upon him, initially forbade the writing down of Hadith for fear of its mingling with the Qur'an. Then, he later permitted it. The organized compilation of the Sunnah began by the command of Omar ibn Abd al-Aziz, may Allah have mercy on him, in the late first century Hijri. Scholars successively gathered the Ahadith with their chains of narration, leading to the emergence of numerous books such as the Two Sahihs, the books of Sunan, Musnad collections, and Musannaf works. The authentic Hadith can be distinguished from the inauthentic thanks to the efforts of Hadith critics who dedicated their lives to discerning it, such as Ahmad ibn Hanbal and Al-Bukhari, all the way to later and contemporary scholars like Al-Albani. A layman should not narrate a Hadith attributed to the Prophet, peace and blessings be upon him, until its authenticity has been verified.