In short, when Jesus spoke, he did so as both man and God. However, his speech was an action of his human nature, not his Divine Nature.
Regarding the passages that seem to imply a subordination of Jesus to the Father, there is no “official” way to interpret them, but any interpretation must, of course, be consistent with the dogma of the perfect equality and consubstantiality (homoousios) of the Persons of the Father and the Son.
Background: Jesus Christ is true God and true man
As other posters have correctly mentioned, the Catholic Church teaches that Jesus Christ is the Divine Person of the Word (or Son), and that, through the Incarnation, he assumed a human nature. (In this, the Catholic Church shares the exact same faith as all the Eastern Orthodox churches). As the Catechism of the Catholic Church (CCC) puts it,
The unique and altogether singular event of the Incarnation of the Son of God does not mean that Jesus Christ is part God and part man, nor does it imply that he is the result of a confused mixture of the divine and the human. He became truly man while remaining truly God. Jesus Christ is true God and true man (no. 464).
A full treatment of dyophysism (the orthodox view that Christ has two natures) is given in numbers 464-483.
What is a “nature”?
A word must be said about the terms “nature” (Greek: physis) and “Person” (Greek: hypostasis) which are borrowed from Aristotelian philosophy.
“Nature” (physis) in Aristotelian philosophy is a synonym for “essence” or “substance” (ousia, or more precisely, to ti en einai), which means “what a thing is simply because it exists”—in other words, a being that is both concrete and independent, as opposed to an accident or a constituent principle. (See Metaphysics 7, 3.)
Aristotle, however, calls essence physis when he considers it as the origin of of a thing’s actions. A man’s human nature is what produces his speech, his movements, and his thoughts—all of the actions (both spiritual and corporeal) characteristic of a man. (See Metaphysics 5, 5.)
Synthetically, “nature” (along with “essence” and “substance”) answers the question “What is it?” but the term emphasizes the role of the substance as origin of action.
Hence, when the Church Fathers asserted that Christ possesses a perfect human nature, they were simply saying that Jesus is man, and possesses every aspect of manhood (a human body and a human soul, with the capacites that go with them: intellect, will, and all the bodily capacities). Moreover, this human nature is what produced all of the actions that are charistic of human beings (human knowledge, human love, speech, laughter, eating, drinking, sleeping, and so on).
Likewise, when they assered that Christ posseses a Divine Nature, they mean that he is also fully God. That Divine Nature, moreover, is the origin of all the actions characteristic of God: creation, redemption, forgiveness of sins, miracles, and so on.
What is “Person” (“Hypostasis”)?
In Greek philosophy, hypostasis is the term that emphasizes the concreteness of a thing. Whereas “nature” and “essence” can sometimes (as a secondary meaning) be used as generic terms—just as even in modern parlance “human nature” can refer to what all human beings have in common—“hypostasis” always refers to the concrete individual. (See Summa theologiae Ia, q. 29, a. 2.)
In Trinitarian theology, the term hypostasis was assumed by the Cappadocian Fathers (Basil the Great, Gregory of Nazianzen, Gregory of Nyssa) to refer to what Western Christians call the Persons of the Trinity.
A Person or Hypostasis is a subject who can refer to himself as “I” and can answer the question “Who?”
It took some time for the Church to understand that the Hypostasis of the Word (or Son) is the very same Hypostasis of Jesus Christ (CCC 468).
When we ask, “What is Jesus Christ?” the answer is, “God and man.” However, when we say “Who is Jesus Christ?” the correct answer is “God the Son, the Second Person of the Trinity.”
Because his human nature is so closely united to his Divine Person—a union we call the hypostatic union, because the principle of unity is the very Person (Hypostasis) of the Word—we can attribute with all properness every one of Jesus’s actions to either one of his natures.
For example, it is perfectly correct to say “Jesus the man created the universe.” Likewise, it is perfectly correct to say “God spoke to the crowds in parables” or even “God died on the Cross on Good Friday.” This is possible because there is only one personal subject in Jesus: the Person (Hypostasis) of the Son.
This common attribution is what the Church Fathers called the communicatio idiomatum (which could be translated as the “communion of properties”).
However (recalling that nature is always the origin of a thing’s actions), Jesus’ actions originate from different natures.
Jesus the man created the universe, certainly, but he did it through his Divine Nature. God spoke and even died, but He did so in His human nature.
Jesus spoke as God and as man
In summary—and responding to the original question—Jesus Christ certainly spoke as both God and man.
He spoke as God, because he is a Divine Person, and that Person is the only personal subject in him. He spoke as man, because he possesses a complete human nature (including human body, human soul, and all the capacities proper to a man).
However, when he spoke, the origin of his speech—what directly produced the speech—was his human nature. Of course what he revealed in that speech came from his Divine Nature.
(Note that the ultimate origin of Christ’s human nature and the actions produced by it is, of course, his Divine Nature, because even Christ’s human nature—not his Divine nature!—was created.)
Regarding passages that seem to imply subordination
But concerning that day and hour no one knows, not even … the Son (Mt. 24:36)
Although there are many possibilites, the most plausible explanation for this saying (in my opinion) is that it was a Semitic hyperbole. In essence, Jesus is emphasizing (for modern audiences, to a ridiculous degree) that under no circumstanes will he reveal the day or the hour of his Second Coming.
Hence, this saying is to be taken figuratively. It is not intended to be a theological assertion about the consubstantiality of the Father and the Son, nor of the degree to which Jesus in his human nature shares in divine knowledge.
In other words, in my opinion, Jesus did know, but refused to reveal it.
In this opinion, I am following that of the of the Ignatius Catholic Study Bible: New Testament. (See the footnote for Matthew 24:36.)
The Father is greater than I (John 14:28)
This could be understood in two plausible ways:
Jesus is referring to his current condition, in which he voluntarily “empties himself” (in the words of Phil. 2:7) by taking on a human nature. He will do greater works when he ascends into heaven. In other words, he is saying, in essence, “The Father is greater than I, as you are seeing me now, in my human nature.” (Keep in mind that in general Jesus chose not to use precise theologial or philosophical language.)
When Jesus says the Father is “greater,” he is referring to the order of the Persons in the Trinity, not their consubstantiality. It is true that the Father is first, being the ultimate origin of the Processions.
As is typical in the Gospel of John, he may very well have meant both things at once.
(Again, I took this interpretation from the commentary on this verse found in the Ignatius Catholic Study Bible.)