Is it to do with the maternal image of care and love or something of the sort? Or did Mary actually do something to deserve her deification?
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The Holy Family of Jesus, Mary, and Joseph has a complex cultural history that's already apparent in the writing of the Gospels in the first century AD. Evangelists Matthew and Luke both provide accounts of the run-up to the Nativity. Matthew, whom scholars generally agree was writing for an audience of Jewish Christians, has the angel foretell Jesus' birth to Joseph in a dream. Luke, whom scholars see as writing for a Gentile Christian community, has the angel visit Mary.
Despite the Christian propensity to harmonize the Gospel accounts of Jesus' life from very early on, several factors united in late antiquity to push Mary to prominence over Joseph. It's worth mentioning that the Crucifixion and Easter were more important than the Nativity (Christmas) in Christianity (technically still are), and Joseph is Sir Not Appearing in the Passion narrative. However, you could easily argue that if Joseph were important, Christians would have emphasized the story that involved him, so that's a bit of a non-starter. And anyway, some later medieval authors had no trouble telling the Nativity story with little role for Joseph.
Probably the biggest factor, though, is that a major point of contention in the early Church, as the basics of "Christian" belief were being hammered out, was the divinity of Jesus. And this question was frequently expressed in terms of Mary: was she the Theotokos or Christotokos, bearer of God or bearer of the messiah? The Mary-Theotokos, Jesus-God camp famously won the day in little-o orthodoxy.
The cultural valuation on Mary over Joseph in the early Middle Ages is perhaps most visible in an odd source--the Quran! The Quranic account of Isa and Maryam (who has two suras named for her!) does not include Joseph at all.
But while Eastern Christianity and, for some reason, Anglo-Saxon England nurture particularly strong devotion (cultus, as in "cult of the saints") to Mary already in the early medieval period, it's really the high=>late Middle Ages in the west that elevate the Virgin Mother to a status that sometimes seems to border on divinity for this very special but very human woman. (In fact, you can trace real concern in later medieval sources that some people were indeed taking Marian devotion too far, but I'm getting ahead of the story here.)
One geopolitical reason for this is the aftermath of the Norman Conquest and the tightening of ecclesiastical and intellectual ties between the insular and continental Church. I mentioned that Anglo-Saxons had a thing for Mary, which endured in the English Church. Vastly Important medieval theologian Anselm of Canterbury (late 11th century) doesn't quite get there himself, but the body of writings surrounding him gives us the first articulations of the Immaculate Conception idea. This is everyone's favorite "Mary hey did you know" fact, but I'll say just in case that the conception here is that of fetus Mary. The point in contention is whether Mary, who according to Christian tradition never sinned during her life, was conceived by her mother also without original sin (hence, immaculate--without stain). It is NOT a question of a virgin conception/birth. Some monasteries (maybe also parishes) are already celebrating it as a feast day, even. Growing ties to the continent result in exchanges of ideas, theology, traditions as ecclesiastics and scholars travel back and forth with increasing regularity, especially from the 12C onward.
The BIG religious development implicated in Mary's rise to become the Queen of Heaven/Empress of Hell in the later Middle Ages, however, is the blossoming of devotion to what scholars call "the humanity of Christ". In Christian theology, Jesus is simultaneously human and divine, but the different natures (let's not refight the ecumenical councils, please) have received different amounts of emphasis at different points in time. The later Middle Ages turn HARD to the humanity of Christ.
This is a complex phenomenon, but to give the factor that is also very relevant for Marian prominence: the 12th-13th centuries is also the time where the Church realizes it needs to and then commits to actually paying attention to the laity. Regular people are seizing religious initiative for themselves on an unprecedented scale. Closeness to Jesus drives and is driven by this trend. Not in the "have you accepted Jesus Christ as your personal Lord and Savior" sense of modern Left Behind evangelicalism, though. While monks, nuns, and itinerant holy people (especially women) seize upon the idea of Christ as their spouse and write some achingly gorgeous and disturbingly erotic poetry about it, a bigger deal on a societal level is the imitation of Christ. People want to/are taught to live a Christ-like life. In pastoral teaching, the emphasis is of course on virtues and avoidance of vice. But it also means the embrace of Christ's humility in taking on human form--his poverty and his suffering.
This also relates closely to evolving theology on salvation, which is after all the whole point of Christianity (we are almost back to Mary, I swear!). Although again, the emphasis is not Jesus the personal savior, later medieval teaching stresses the Passion and crucifixion as Jesus' suffering and redemption of people. The suffering--the very human suffering--is crucial to late medieval popular theology of salvation. And this is where Mary rises to such a critical role in later medieval Christianity.
As the vessel through whom Jesus enters the world, Mary is physically the mediator between God and humanity. The late medieval, and especially late medieval Christian, turn towards materiality (WEEKLY THEME SHOUT OUT) and crude embodiment only highlights the viscerality of this connection further. Through it, Mary takes on a privileged role, the most privileged role, in a late medieval religion obsessed with the quest for salvation. Mary is mediatrix: the ultimate negotiator between the human soul and God in light of the Last Things. She is the voice for mercy at the end of all things. Medieval pastors try REALLY hard to stress that Mary herself has no salvific power. It mostly works. Most of the time.
Mary's role in the popular Christian imagination is reinforced in the 14th and 15 centuries for an additional, odd reason: petty theological bickering. The two major preaching/pastoral orders of the late Middle Ages were the Dominicans and the Franciscans, both founded in the early 13th century. They were rivals for ecclesiastical favor, lay attention, and all sources for money, of course, but they could also be theological rivals.
The Franciscans and Dominicans both nurtured a "special devotion" to Mary, but from the 14th century, it was one with a crucial theological difference. Mary's growing fama was triggering a landslide of public desire to celebrate the Feast of the Immaculate Conception. The major theological figures of the mendicant orders' early history (13th century) both nurtured that "special devotion" to Mary, but both opposed the Immaculate Conception.
The Dominicans of the 14th century, for whom Thomas Aquinas was still The Guy, stuck with his argument that Mary could not have been conceived without sin, because that would make Jesus' salvation-of-humanity less than total, ergo less than perfect. The Franciscans snorted and said, "Bonaventure was great, but we are not afraid." 14C Franciscan theologians argued that Mary's conception without sin made Christ's redemption even more perfect, because he had redeemed her before she was even conceived.
Well, in competition for medieval audiences, Dominicans and Franciscans were at each other's throats over this. So you find LOTS of attention to Mary in sermons that subtly or not so subtly argue for/against the Immaculate Conception. The theological fight served to emphasize Mary's role to the public even further.
By 1400, increasing vernacular literacy among the urban laity combined with skyrocketing public religiosity (the coming 15th century is known today as the "most church minded and devout" period of medieval history!) puts Nativity, Passion, and overall Life of Christ narratives in the hands and hearts of a growing number of Christians. The books encourage them to meditatively/imaginatively insert themselves into the stories recounted. Frequently, Mary serves as a vehicle for this! Readers are urged to nurse the infant Christ with Mary, to wipe his tears with her/their hair, to kiss his feet. They watch helplessly with Mary as the adult Christ received his crown of thorns; they hold her hand at the foot of the cross. Joseph is generally absent in these texts.
However...
Follow up question, is there any information about how marriages were consumated during that time period? It always seemed a little strange to me that they were married but she was still a virgin.
The veneration of Mary has a lot to do with her title as Θεοτόκος (Theotokos), which means Bearer of God. We see this term in usage even within the Early Church - specifically, during the Ante-Nicene, before the Council of Nicaea.^1 ^2
This title was later formally affirmed in 431 CE by the Third Ecumenical Council, also known as the Council of Ephesus.^3 Though most people are likely familiar with the phrase "Holy Mary, Mother of God", the use of this title was an important theological issue to the Early Church.
After this council declared those who denied theotokos anathema, the Church began heavy theological examination of the consequences of this. This study of the figure of Mary is often called Mariology, if you ever want to investigate further for yourself.
Aside from the Marian apparitions and the miracles attributed to her, theotokos itself holds important implications for Christian doctrine. If Mary is the birthplace of Christ, and Christ is free from original sin - stated in the Council of Trent, then Mary cannot be soiled by original sin. Mary isn't venerated because she is the mother of Christ per se, but because she is the mother of Christ.
There isn't nearly as much literature on Joseph as there is on Mary, at least from what I've read. However, certainly present is the idea that Jesus isn't related to Joseph in the same way Jesus was to Mary, by blood.
Finally, the father of Christ was never anticipated like the mother of Christ was.
Therefore the Lord himself will give you a sign. Behold, the virgin shall conceive and bear a son, and shall call his name Immanuel. Isaiah 7:14 (ESV)
TL;DR: Mary's veneration is largely because of:
1 Madeleine Pelner Cosman, Linda Gale Jones. Handbook to Life in the Medieval World, 3-Volume Set, Volumes 1-3. New York; Facts on File, 2008. p. 331.
2 "Dionysius Of Alexandria". earlychristianwritings.com. n.p., 2017. Web. 6 Jan. 2017.
3 "NPNF2-14. The Seven Ecumenical Councils - Christian Classics Ethereal Library". Ccel.org. n. p., 2017. Web. 6 Jan. 2017.
4 Great Events in Religion: An Encyclopedia of Pivotal Events in Religious. Edited by Florin Curta, Andrew Holt. ABC-CLIO. p. 273.
5 Jaroslav Pelikan. Mary Through the Centuries: Her Place in the History of Culture. London; Yale University Press. 1996. p. 8-9.
Just a clarification. As far as Catholics go, Mary is not deified, she is venerated.
"You ask about the differences between veneration and worship. The technical terms that are used seem appropriate for the explanation. The chief ones are dulia, hyperdulia, and latria. Dulia is a Greek term meaning the veneration or homage, different in nature and degree from that given to God, that is paid to the saints. It includes, for example, honoring the saints and seeking their intercession with God. Related to dulia is Hyperdulia, the special veneration accorded the Blessed Virgin Mary because of her unique role in the mystery of Redemption, her exceptional gifts of grace from God, and her pre eminence among the saints. Hyperdulia is not adoration; only God is adored. Such adoration reserved exclusively for God is termed latria, a Greek-rooted Latin term that refers to that form of praise and worship due to God alone. " [Source]
Now as for Mary, the Catholic Church teaches that she is "Mother of Christ, Mother of the Church" [Source] and held to be worthy of great devotion:
"The Church's devotion to the Blessed Virgin is an intrinsic element of Christian worship. The honor which the Church has always and everywhere shown to the Mother of the Lord, from the blessing with which Elizabeth greeted Mary (cf. Lk. 1:42-45) right up to the expressions of praise and petition used today, is a very strong witness to the Church's norm of prayer and an invitation to become more deeply conscious of her norm of faith. And the converse is likewise true. The Church's norm of faith requires that her norm of prayer should everywhere blossom forth with regard to the Mother of Christ. Such devotion to the Blessed Virgin is firmly rooted in the revealed word and has solid dogmatic foundations. It is based on the singular dignity of Mary, "Mother of the Son of God, and therefore beloved daughter of the Father and Temple of the Holy Spirit-Mary, who, because of this extraordinary grace, is far greater than any other creature on earth or in heaven."(119) This devotion takes into account the part she played at decisive moments in the history of the salvation which her Son accomplished, and her holiness, already full at her Immaculate Conception yet increasing all the time as she obeyed the will of the Father and accepted the path of suffering (cf. Lk. 2:34-35, 41-52; Jn. 19:25-27), growing constantly in faith, hope and charity. Devotion to Mary recalls too her mission and the special position she holds within the People of God, of which she is the preeminent member, a shining example and the loving Mother; it recalls her unceasing and efficacious intercession which, although she is assumed into heaven, draws her close to those who ask her help, including those who do not realize that they are her children. It recalls Mary's glory which ennobles the whole of mankind, as the outstanding phrase of Dante recalls: "You have so ennobled human nature that its very Creator did not disdain to share in it."(120) Mary, in fact, is one of our race, a true daughter of Eve-though free of that mother's sin-and truly our sister, who as a poor and humble woman fully shared our lot." [Source]
So where does that leave St. Joseph?
Joseph was "a just man". This praise bestowed by the Holy Ghost, and the privilege of having been chosen by God to be the foster-father of Jesus and the spouse of the Virgin Mother, are the foundations of the honour paid to St. Joseph by the Church. So well-grounded are these foundations that it is not a little surprising that the cult of St. Joseph was so slow in winning recognition. [Source]
His veneration is recorded in the East as early as the fourth century but not in the West for many centuries later. The growth in the cult of the saint in the West is slow and steady, and fully encompassed in Catholic tradition, after the ninth century.
Pope Leo XIII promulgated the Encyclical, Quamquam pluries on the devotion to St. Joseph in 1889 which outlines Josephology and was proclaimed in the context of Joseph as a patron of laborers during the rise of industrial capitalism. Saint John Paul II published the Apostolic Exhortation, Redemptoris custos in 1989, which positions St. Joseph within the context of redemption history.
It's tough to compare the two, Mary and Joseph. Mary is singularly regarded but even Mariology is a field that as been under development and refinement since the early Church. The virgin birth of Jesus was declared a dogma at the First Council of Nicea in 325 but the Dogma of the Immaculate Conception (Mary was conceived free from the stain of original sin) wasn't proclaimed until 1854.
So, the Catholic answer to your questions are:
Q: Why is the mother of Jesus deified but his father is almost not mentioned?
A: She is not deified but venerated. The father of Jesus is God the Father who is worshipped above all things. The foster-father of Jesus, St. Joseph, is venerated and called, among other things, Protector of the Church.
Q: Is it to do with the maternal image of care and love or something of the sort?
A: Theologically speaking, it's something very different than "maternal image of care and love".
Q: Or did Mary actually do something to deserve her deification?
A: Why is Mary worthy of hyperdulia? For her unique role as the Mother of God.