In Plato's Apology, Socrates makes a crucial distinction between human and divine wisdom. Divine wisdom is, essentially, complete and certain knowledge of everything. Basically, to Socrates "divine wisdom" constitutes what we would call omniscience. Human wisdom is far more limited, and while Socrates does not give an overarching definition of it, he does provide four clear examples of what he himself claims to know with his own "human wisdom":
(1) First of all, and most famously of all, Socrates does claim to be able to recognize, with great certainty, ignorance for what it is, both in himself and in others.
(2) Second of all Socrates shows no ambiguity concerning his ethical stance. In particular Socrates is certain that we should always act in accordance with virtue and piety, without, however, claiming absolute knowledge of what virtue and piety are.
(3) Thirdly Socrates claims to have had repeated experiences of direct communication with the divine, by way of his famous "sign". Socrates shows no skepticism whatsoever concerning either the meaning of this "sign" when it is given to him, nor of it's divine origin.
(4) Finally, as a result of his "philosophical mission" Socrates also claims to have demonstrated, at least to his own satisfaction, the true meaning of the oracular pronouncenment that "of all men Socrates is most wise".
Obviously then, the position that Socrates takes in the Apology cannot be reduced to only the claim to "know that I do not know", and nothing more. Moreover, such a characterization of Socrates is highly misleading, since he definitely claims much more than that. Such an oversimplification doesn't even accurately portray the first kind of knowledge that Socrates claims (according to the list above), because he is no less insistent about his ability to recognize, and expose, the ignorance of others than he is with respect to his own ignorance.
Of all these four claims I believe the last one to be the most significant. When Socrates first heard, from his close friend Chaerophon, what the oracle at Delphi had said: that Socrates was the wisest of men, he simply could not believe it. But he did not discount it, much less ignore it. He felt an obligation to get at the truth of what the oracle was saying. Socrates appears to have taken the typical attitude of any pious Pagan: that oracular sayings are mysterious and difficult to understand, but they are genuine messages from the Divine and they are true if one understands their meaning.
In fact, Socrates does nothing less than spend the rest of his life following where the oracle has led him, and from that course he refuses to deviate even if it costs him his life, which it does.