If we can speak about the contradiction between the two passages, the essence of it would be in that the second quotation (2 Tim 3:16) makes the very text of the Holy Writing a product, so to say, of the breath and inspiration of the Holy Spirit, whereas the first quotation (2 Cor 3:6) warns about not following the text, for it may "kill", while "Spirit" will vivify, so how can something which comes from the Spirit (the text) kill at the same time?
But this contradiction is only apparent and not essential at all: the first quotation does not say that, for example, the 10 commandments are not inspired by the Spirit, but that if they are not read by the aid of the same Spirit, even they may become harmful. For instance, if a man fulfills a letter of the commandment "you shall not commit adultery" and just does not go to other women, and yet not paying enough attention to his wife and not cultivating conjugal love, he still essentially commits adultery, but since the letter of the commandment is fulfilled, he does not notice it and thus the letter of commandment "kills" the possibility of development of the conjugal love in fear and service of God.
Thus, to see the essence of any written utterance one should consult, so to say, and interact with the same Spirit, through whom this utterance was inspired, with full participation of one's conscience and intellectual-cognitive powers, for only so it is possible to interact with the Spirit. Now, the intensity of the presence of the Spirit became unimaginably greater for humanity after the Pentecost, for Jesus, as He promised, sent the Spirit to His disciples as "another Comforter" (John 14:16) ("another" with a reference to Himself, for He was Comforter of His disciples while being with them, but now another Comforter equal to Him would guide them to conversion of both Jews and gentiles), thus the text of the OT should be now read in the light of this new intensive presence.
That is why, Jesus does not change the commandments of the OT, but to fulfill (πληρῶσαι) which means to bring them to perfection, that is to say interpret them with the new profundity and depth of the spiritual understanding: thus, "you shall not kill" is now elevated into its spiritual significance "hatred is already killing" (for hatred is a motivational foundation of actual killing and in a sense itself an actual killing of somebody in one's heart and soul), or "you shall not commit adultery" becomes now "you shall not look with a lustful eye at anybody" (which again deals not with action but with its fundament, a motivation and inner disposition).
And the same deepening about everything said in the Old Testament, for the Spirit who knows all depths of God (1 Cor. 2:10) the epistemological equality denoting here also the ontological equality. Moreover, also the New Testament should be read by synergic interaction with Spirit, for without it one may actually castrate oneself and not perceive a profound metaphor in the Matt. 19:12.
This deepening of understanding is so wonderful in the history of Christian theology when fathers give very profound, paradoxical interpretations of the Scriptures, St Augustine in his "De doctrina christiana" even conjectures that Christian interpreters would understand and interpret the inspired utterances of prophets at a more profound level than understood by the prophets themselves, for the Spirit in a dynamical process reveals to them, the Christian theologians, more than He revealed to the prophets hundred and thousand years before the advent of Jesus. And more than that, it can be already spiritually hazardous and damaging to read the OT and extract guiding spiritual lessons from it without seeing the text of the OT in the light of the perfect revelation of the New Testament, it is like a real soldier turns to wooden swords of children who play war, for the OT is only a prepaedeutic for the New Testament (Gal.3:24). Thus, St. Basil, for instance, urges beginner Christians first to read deeply the New Testament and hone their consciences in its light, and only then turn to the Old Testament without spiritual peril.
