What is the difference between a "fabricated" (mawḍūʿ) ḥadīth and a "weak" (ḍaʿīf) ḥadīth? And how was each chain of narration judged?

A weak Hadith is one that has lost one or more of the conditions for an accepted Hadith. Its weakness is divided into two categories: that which is due to a lack of connection (such as al-Mu'allaq, al-Munqati', al-Mu'dal, al-Mursal, and al-Mudallas), and that which is due to a damaging flaw in the narrator (such as al-Majhul, al-Layyin, al-Maqlub, al-Mushaff, al-Mudraj, al-Shadhdh, al-Mu'allal, al-Mudtarib, al-Munkar, and al-Mawdu'). The fabricated Hadith is that which is invented and falsely attributed to the Prophet, peace and blessings be upon him, whether it has a chain of narration or not.