The Protogonos, the Eudemian, the Rhapsodic and the Hieronyman theogonies are all reconstructed, discussed and compared in ML West (1983)'s 'The Orphic Poems'. Unfortunately, I do not have access to this work, so my answer is based solely on reviews, commentaries and criticisms of this book (such as Brisson, 1985 or Betegh, 2004).
The main difference seems to be in the primordial succession:
- In the Eudemian theogony, all starts with the Night, which lays an Egg from which Phanes/Protogonos arises.
- In the Rhapsodic theogony, it starts with Chronos ('Unageing Time', different from Kronos, Zeus' father) who gives birth to Ether and Chaos, and then lay the egg from which Phanes/Protogonos arises.
- In the Hieronyman theogony, the egg arises from soil (more specifically 'the matter out of which earth was coagulated') and water, and it is 'Unageing Time' Kronos which arises from it, and gives birth to Ether, Chaos and Erebus. Then Kronos lay a new egg in Chaos, from which arises Protogonos.
- In the Derveni papyrus, a. k. a. the 'Protogonos' theogony, the Night lays the egg from which Protogonos arises, he then give birth to Ouranos & Gaia, which give birth to Kronos, himself father of Zeus who end up swallowing the primordial egg of Protogonos and recreating the Universe in the process.
But there are other differences, notably in the treatment of Dionysos:
In the Eudemian and Rhapsodic theogonies, Dionysos is dismembered and cooked by the Titans before Zeus struck them with lightning (mankind then arises from the soot, and Dionysos is resurrected from his preserved heart).
The Derveni Papyrus being fragmentary, the story stops without having mentioned him.
The Hieronyman theogony do not mention Dionysos being eaten by the Titans in neither source it is known from (Damascius and Athenagoras), despite the latter describing the war on the Titans, which would imply that this story really isn't part of that theogony.