How can John the Baptist be the greatest man born of woman, when Jesus was also born of woman?

John the Baptist was "Man born of woman."
Jesus, God's Son, was "God Born of Woman."

Jesus was God's Son, the Living word of God born of or delivered by woman. John's Gospel Chapter 1 tells us that

In the beginning was the word and the word was with God and the word was God...... Jesus "came" unto His own but His own received Him not.

In the introduction to his letter to the Romans, Paul says this:

³Concerning his Son Jesus Christ our Lord, which was made of the seed of David according to the flesh;
-- Romans 1:3 (KJV)

The Greek here is: τοῦ ^{the one} γενομένου ^{upon being made} ἐκ ^from σπέρματος ^{of the seed} Δαυὶδ ^{of David}

Also, in his letter to the church at Galatia, Paul says this:

⁴But when the fullness of the time was come, God sent forth his Son, made of a woman, made under the law,
-- Galatians 4:4 (KJV)

Again, the Greek here is: γενόμενον ^{upon being made} ἐκ ^from γυναικός ^{of woman}

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The passage quoted by the OP contains this:

¹¹Verily I say unto you, Among them that are born of women there hath not risen a greater than John the Baptist: notwithstanding he that is least in the kingdom of heaven is greater than he.
-- Matthew 11:11 (KJV)

The Greek here is: ἐν ^among γεννητοῖς ^{born ones} γυναικῶν ^{of women}

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So, Jesus says John was γεννητοῖς ^born of a woman, but Paul says Jesus was γενόμενον ^{made of} a woman. What's the difference?

Clearly, being "born of a woman" relates to conception by means of human sexual union, i.e. the process that gives rise to the fertilisation of an ovum by a sperm cell. However, Jesus wasn't born in this way:

³⁵And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God.
-- Luke 1:35 (KJV)

Apart from those who believe that non-corporeal beings have reproductive bits and pieces with which they can mate with corporeal beings, then this depicts what Paul referred to as being "made of a woman", i.e. being knit together in the womb of a woman.

For those who are inclined to imagine sexual union here, Paul strongly refutes such a notion when he writes that Jesus was "made under the law" (Galatians 4:4, above), for when Joseph was contemplating "putting away" (divorcing) his betrothed, the Angel said to Him:

²⁰... Joseph, thou son of David, fear not to take unto thee Mary thy wife; for that which is conceived in her is of the Holy Ghost.
-- Matthew 1:20 (KJV)

It would have broken the law for Mary to have been taken "sexually" and then given to another without a certificate of divorce. Countering the notion of "sexual union" was clearly what Paul had in mind when he wrote that Jesus was "made under the law", else he would not have referred to it.

John records Jesus words:

⁶³It is the spirit that quickeneth; the flesh profiteth nothing: the words that I speak unto you, they are spirit, and they are life.
-- John 6:63 (KJV)

The spirit is what moves a body and gives it life. Those "born of women" have a spirit that is separate from God. The Spirit of God can coexist with the spirit of man to endeavour to move him counter to his fleshly inclinations, but He will not extinguish the freedom of his spirit to do so. Luke records:

¹⁵For he shall be great in the sight of the Lord, and shall drink neither wine nor strong drink; and he shall be filled with the Holy Ghost, even from his mother's womb.
-- Luke 1:15 (KJV)

So, the Holy Spirit coexisted with John's spirit from the moment he entered into the world.

However, the manner of Jesus birth had to be such that the Spirit of God alone was the power that moved him. The bag of bones (which profiteth nothing) that bore the name Jesus, had a beginning, but the Spirit that moved the bag of bones was the eternal Spirit of God alone. Jesus was free to express the agony and anguish of his flesh through earthly trials, but it was God Himself who experienced the anguish and agony and rejection, not the spirit of a man with whom the Holy Spirit coexisted.

Conclusion

When Jesus said, "among those born of women there has arisen no one greater than John the Baptist", he was fully cognisant of the miraculous nature of His own birth. Jesus was not born of sexual union, but of "the power of the highest" that literally knit him together in the womb of his mother - a holy gamete formed from the genetic attributes of both Joseph and Mary.

It is only at our end of the timeline that we can fully comprehend what God had done.

I really like enegue's answer.

I would only add that the distinction being drawn by Jesus

11 Truly, I say to you, among those born of women there has arisen no one greater than John the Baptist. Yet the one who is least in the kingdom of heaven is greater than he. 12 From the days of John the Baptist until now the kingdom of heaven has suffered violence,[d] and the violent take it by force. 13 For all the Prophets and the Law prophesied until John, 14 and if you are willing to accept it, he is Elijah who is to come. (Matthew 11 ESV)

is between those born of the flesh and those born of the Spirit.

As in Jesus' conversation with Nicodemus a distinction exists between those of purely natural birth and those who are born from above. Jesus, while firstborn from the dead and preeminent in all things, was, at the time, having taken the lowest place in the flesh, the last which shall be first. He is the greatest in the Kingdom of Heaven and is therefore (as are all who are born again by faith in him) excluded from the category of "those born of women".

How can John the Baptist be the greatest man born of woman, when Jesus was also born of woman?

11 Truly, I say to you, among those born of women there has arisen no one greater than John the Baptist. Yet the one who is least in the kingdom of heaven is greater than he. 12 From the days of John the Baptist until now the kingdom of heaven has suffered violence,[d] and the violent take it by force. 13 For all the Prophets and the Law prophesied until John, 14 and if you are willing to accept it, he is Elijah who is to come. (Matthew 11 ESV)

First of all, it seems logical that Our Lord was not speaking of himself. True, Our Lord was born of a woman also. But Jesus was likewise conceived by the power of the Holy Spirit and thus was not conceived in the same manner that the rest of mankind is.

Seeing that you quoted Pope Francis, I think it good to give a more Catholic perspective to your question.

There seems to be no official statements on the subject of Matthew 11:11, at least not at the level of involving papal infallibility. Nevertheless, there are some Catholic writers and Doctors of the Church who have written on the subject of St. John the Baptist as being the greatest born of women, yet remaining the below the least in the kingdom of heaven.

We see in the Psalms that man was made little less than the angels (Psalm 8:4-6).

St. Thomas Aquinas quotes the passage of Matthew 11:11 several times in his Summa Theologica for various questions. We must also keep in mind that at the moment Our Lord spoke these words about St John, mankind had not yet been admitted into the kingdom of heaven because Jesus' sacrifice on the Cross had not yet occurred.

Now St. Thomas tells us in one of his questions (Whether there are several orders in one hierarchy?): "The inferior angel is superior to the highest man of our hierarchy, according to the words, "He that is the lesser in the kingdom of heaven, is greater than he"---namely, John the Baptist, than whom "there hath not risen a greater among them that are born of women" (Mat. 11:11). Hence the lesser angel of the heavenly hierarchy can not only cleanse, but also enlighten and perfect, and in a higher way than can the orders of our hierarchy. Thus the heavenly orders are not distinguished by reason of these, but by reason of other different acts.'

Thus one can conclude that Our Lord spoke about St John in comparison with the angels who art in heaven.

In another question (whether Moses was the greatest of the Prophets), St. Thomas stated: "Further, it is written (Matthew 11:11) that "there hath not risen, among them that are born of women, a greater than John the Baptist." Therefore Moses was not greater than all the prophets."

As Catholics we believe that, by a divine prerogative from God, Mary was conceived with no stain of original sin. It is not impossible that St. John, although he was not conceived immaculately may have been purified while still in the womb of St Elizabeth! Although not dogma, the Church has not yet pronounced on this subject.

"There is a solid tradition in the Church that says St. John the Baptist was purified of original sin shortly after he was conceived, while still in the womb of St. Elizabeth. So, this episode of the Gospel referring to the child in the womb hearing Our Lady’s voice, understanding her words and loving her is completely credible." - Professor Plino Correa de Olivra

Here is how Wikipedia puts it:

"Some Catholics have held to a belief that John the Baptist never sinned, though this has never been a point of doctrine and is not binding in belief upon any adherent as is the sinlessness of Mary. In her Treatise of Prayer, Saint Catherine of Siena includes a brief altercation with the Devil regarding her fight due to the Devil attempting to lure her with vanity and flattery. Speaking in the first person, Saint Catherine of Siena responds to the Devil with the following words:

...humiliation of yourself, and you answered the Devil with these words: 'Wretch that I am! John the Baptist never sinned and was sanctified in his mother's womb. And I have committed so many sins..." — Catherine of Siena, , A Treatise of Prayer, 1370.[85][86]" St Catherine of Sienna was declared a Doctor of the Church on October 3, 1970 by Pope Paul VI.

The Catholic Encyclopedia has this to say:

"Now during the sixth month, the Annunciation had taken place, and, as Mary had heard from the angel the fact of her cousin's conceiving, she went "with haste" to congratulate her. "And it came to pass, that when Elizabeth heard the salutation of Mary, the infant" — filled, like the mother, with the Holy Ghost — "leaped for joy in her womb", as if to acknowledge the presence of his Lord. Then was accomplished the prophetic utterance of the angel that the child should "be filled with the Holy Ghost even from his mother's womb". Now, as the presence of any sin whatever is incompatible with the indwelling of the Holy Ghost in the soul, it follows that at this moment John was cleansed from the stain of original sin. When "Elizabeth's full time of being delivered was come. . .she brought forth a son" (1:57); and "on the eighth day they came to circumcise the child, and they called him by his father's name Zachary. And his mother answering, said: Not so, but he shall be called John. And they said to her: There is none of thy kindred that is called by this name. And they made sign to his father, how he would have him called. And demanding a writing table, he wrote, saying: John is his name. And they all wondered" (1:59-63). They were not aware that no better name could be applied (John, Hebrew; Jehohanan, i.e. "Jahweh hath mercy") to him who, as his father prophesied, was to "go before the face of the Lord to prepare his ways; to give knowledge of salvation to his people, unto remission of their sins: through the bowels of the mercy of our God" (1:76-78). Moreover, all these events, to wit, a child born to an aged couple, Zachary's sudden dumbness, his equally sudden recovery of speech, his astounding utterance, might justly strike with wonderment the assembled neighbours; these could hardly help asking: "What an one, think ye, shall this child be?" (1:66)." - New Advent.

Now if St. John was in fact purified while in the womb of his mother he would the greatest of those born of women. Mary's immaculate conception still puts her above St. John in the eyes of the Church, by a divine unique privilege accorded to the Mother of Jesus. This however is a separate question all together.