How can John the Baptist be the greatest man born of woman, when Jesus was also born of woman? Pope Francis tweeted today:


  Both John the Baptist, who is the greatest man born of woman, and the
  Son of God have chosen the path of humiliation. God shows this path to
  Christians so they can move forward. One cannot be humble without
  having suffered humiliation. - #HomilySantaMarta


This definitely connects with (and may be based on) the following passage from Scripture:


  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)


In these verses, Jesus does clearly say that there is no one born of women who is greater than John the Baptist. He also places John in the position of Elijah (who was prophesied to return, as far as I know). It is easy to understand why John the Baptist is great, and there is no one greater than him except for Jesus. But Jesus does not exclude himself in this statement, which surprises me.

I would automatically assume that Jesus is greater than John the Baptist. Jesus is the Messiah, the only begotten Son of God, the Lord and Savior of Christians. Jesus is also born of woman (the Virgin Mary). How can John the Baptist be the greatest man born of woman, when Jesus Christ was also born of a woman?
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

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

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.