In Oedipus Rex by Sophocles, Oedipus is the King of Thebes and Teiresias a prophet. In character both are intelligent, stubborn, and important within their society.
The first difference is in their roles. Teiresias is a prophet who is devoted entirely to his role as the spokesperson of the gods; he is neither ruler nor ruled within human society but instead independent, accountable only to the gods.
Oedipus, on the other hand, is a ruler appointed by popular acclaim, and is responsible for human affairs. Because, until we discover his true birth, he holds his position by popular acclaim, his legitimacy is temporary and consensual rather than divinely authorized.
At the end of the play, when Oedipus gives up his rulership and blinds himself to wander, he becomes much like Teiresias.