Greek choruses play a vital role within the structure of Greek drama and serve as one of its primary defining features (one which also helps distinguish it from modern drama). Indeed, the primary role was usually to provide commentary and exposition for the audience, and, in sharp contrast to the named characters of the play, Greek choruses rarely had a strong sense of identity, personality, or motivation. I use the term rarely because, as can be seen in Eumenides, there are definite exceptions to this rule. More than anything else, a chorus tends to represent a unified group, and it acts and is defined by its collective nature.
Indeed, most of the traditional hallmarks and roles of the Greek chorus can be seen present in both Oedipus Rex and Eumenides. In Sophocles's play, the chorus represents the elders of Thebes, whereas Aeschylus's chorus represents the Erinyes (in both cases, we see the same collective dynamic at play), and in both cases, you can see them fulfilling the chorus's traditional role of providing exposition and commentary.
Yet, even as Aeschylus's play is the older of the two, in certain respects his use of the chorus can be considered more daring and dramatically innovative than Sophocles's, given the degree to which Aeschylus's chorus is dramatically integrated within the play. In Oedipus Rex, the chorus does not really have a vital role within the drama of the play; in fact, were you to remove it entirely from the action of Oedipus Rex, the overall structure and plot would remain intact. The same cannot be said with Eumenides.
Eumenides follows Orestes as he is being pursued by the Erinyes (who seek revenge for the murder of his mother) and later put on trial. In this sense, the chorus actually has a vital role within the drama itself, both thematically and dramatically. You can remove the Theban elders from Oedipus Rex and the overall dramatic narrative remains intact. With Eumenides, however, if you were to remove the chorus, everything collapses.