I'm reading about early Greek rhetoric (Greek Rhetoric Before Aristolte by Richard Enos) and it appears, to me, that aoidoi were singers of Greek epic poems and that they were probably responsible for "composing" the Iliad and the Odyssey. Rhapsodes appear to be very similar. The term rhapsode was not used in early writings, however; but Plato uses the term in his Ion.
I might have found an answer. Enos states: "These predecessors [aoidoi] to Rhapsodes were developing and employing systems of oral discourse even before the Homeric age....What Plato was doing [in Ion], however, was substituting a contemporary Attic Greek term for an outdated Homeric Greek term and providing an important clue to the development of the term "rhapsode" and an indication of the need to examine its Homeric origin" (p.32-33).
"In brief, it would be more precise to consider these Homeric chanters as 'pre-Rhapsodes'; we should qualify the meaning within the context rather than obscure what is essentially the same phenomenon with various labels" (p. 33).
One distinction appears to be that rhapsodes were from a literate age and relied on written texts to help them perform, while aoidoi were from a pre-literate age. This distinction seems to be exaggerated by scholars to demonstrate that rhetoric developed with written texts, whereas Enos (and others) argue "that aoidoi such as Homer and Hesiod were the first Hellenic thinkers to advance formalized systems for both presenting and understanding oral literature--an activity that ought to be recognized as contributing to the evolution of rhetoric as a discipline" (p. 34-35).
I think that while many here might think, "well, duh!" it's important to situate the book in terms of a tradition of thought that believed that only "genius" (p. 21) and not formalized systems of rhetoric, could inspire the creation of works like the Iliad and Odyssey. This bias goes back to Plato. While some ancients, like Quintilian thought highly of "the composing techniques of Homer," most thought that "Homer predated rhetoric as a discipline and [that] Homeric discourse was thought to be a nonrational approach to expression" (p. 21).