What is the difference between a supererogatory prayer (nafilah) and a sunnah prayer, and are sunnah prayers like Tarawih and Qiyam al-Layl accepted if there are missed obligatory prayers to make up?

The Sunnah, in the parlance of the Usuliyyun (jurists specializing in the principles of Islamic jurisprudence), is that for which one is rewarded for performing and not punished for neglecting. The intent behind the saying of Al-Siddiq, "Indeed, Allah does not accept the supererogatory until the obligatory is fulfilled," is that a person is not rewarded for the supererogatory if the obligatory is deficient. Rather, the supererogatory makes up for and completes the deficiency in the obligatory. This is clarified by the Hadith: "Indeed, the first thing a servant will be held accountable for on the Day of Resurrection is his prayer. If it is found complete, it will be recorded as complete. If there was any deficiency in it, [Allah] will say: 'Look, do you find for him any voluntary prayers?' If he has voluntary prayers, the obligatory will be completed from his voluntary prayers. Then, the same will be done for all other obligatory deeds." Therefore, supererogatory acts are only truly effective after the completion of the obligatory. If a servant does not perform the obligatory duties as commanded, he will not achieve the intended purpose of the supererogatory acts.

As for the Sunan, such as Taraweeh and other regular supererogatory prayers (Sunan Rawatib), their performance is recommended because they were legislated for further drawing near to Allah.