Nippur is the clearest ancient case of religious authority deliberately held apart from political power.
It was never a capital. No dynasty of Nippur ruled Sumer. What it held was Enlil, and Enlil's temple the Ekur, the mountain house, was where the assembly of the gods was held to meet and where kingship was granted. A ruler who wanted the title king of Sumer and Akkad needed Enlil at Nippur to have given it to him, and that could not be arranged anywhere else or manufactured at home.
The practical consequence was unusual. Cities that fought each other constantly tended not to sack Nippur, because destroying it would have destroyed the thing that made their own victories legitimate. Successive conquerors instead competed to endow it, rebuild its temples and record their piety there. It was, in effect, protected by being useful to everyone.
The second reason it matters is accidental. Nippur was the great scribal centre, and its schoolrooms, the edubba, are where students copied texts as exercises. Because those exercises were thrown out as rubbish and buried, they survived. The large majority of what can be read of Sumerian literature comes from Nippur: the city laments, the hymns, the debate poems, the wisdom texts, much of the Gilgamesh material, and the flood account. Royal libraries preserved what kings wanted kept; Nippur preserved what children were made to write out, which turns out to be a far better sample.
American expeditions from the University of Pennsylvania worked there in four campaigns between 1889 and 1900 and removed tens of thousands of tablets, which is why the modern study of Sumerian was effectively founded in Philadelphia and Istanbul rather than in Iraq. Among the finds is a clay tablet carrying a measured plan of the city with its walls, canal and gates marked, the oldest known city map.