The Greek Catholics originated in Eastern European on the boundary between Roman Catholic and Orthodox lands. Their church is a religious hybrid, combining Russian Orthodox liturgy with allegiance to the Pope. John Paul II’s Ukrainian mother was Greek Catholic and some believe he was elected partly the hope of unifying the Church.
Situated between Roman Catholic Poland and Orthodox Russia are Belarus and Ukraine. In the 16th century this area came to be ruled by the Polish King. Sigismund III wanted to catholicise his Orthodox subjects and made them an offer. They could retain their married priests and Orthodox liturgy ― and no longer be second-class subjects ― if they would readjust their theology and come under the Pope.  Those who accepted formed new hybrid churches which were called “Uniate” churches from their 1596 union with Rome. Similar unions followed in neighbouring countries. Today these churches are known as “Greek Catholic”.
By the late 19th century Polish rule had ended and much of the area was controlled by the Russian Czar. He, of course, wanted his subjects to be Orthodox and tried to disperse the Greek Catholics. Among these was a girl called Emilia Kacharovska who fled with her family from her home in the Ukraine to the safety of Catholic Poland.  In 1978 her son was elected pope.
Many Slavophiles believe that that Pope John Paul II was probably chosen for the papacy for two reasons: (1) to help bring down the Iron Curtain and restore Christianity to Europe, (2) bridge the divide between Eastern-Orthodox and Western-Latin Europe. The fact that he had both Latin rite and Greek-Catholic parent made him a good candidate for the second task. 
Today, after reconversions, shifting national boundaries and resettlement, most of the Greek Catholics live in Ukraine, in Romania and in Slovakia. Because the Greek Catholics live in traditonally Orthodox lands and their liturgy is reassuringly familiar to their neighbours, the Orthodox hierarchy worries about these Greek Catholics poaching on their territory. In 1993, at Orthodox insistence, the Catholic Church renounced “uniatism”.  Yet ten years later the Patriarch of Moscow and All Russia refused to agree to a visit by Pope John Paul II. He claimed that after the Pope’s trip to the Ukraine in 2001 the Greek Catholics in western Ukraine, who recognize the Pope’s supremacy, “increased their expansion into traditionally Orthodox” areas. 
Whether or not the papal visit was a significant factor, the expansion of the Greek Catholics is undeniable. The apostolic nuncio to Ukraine said, “Greek-Catholics simply do a better job of evangelizing than do most Orthodox.” 
In addition to their Orthodox liturgy, performed in the Slavonic language, rather than Latin,  Greek Catholics retain their own code of Canon Law. However, like other Catholics, they recognise the pope, accept the Catechism of the Catholic Church, are subject to the Congregation for the Doctrine of the Faith and are included in their countries’ concordats.
1. “Pope: Study the Eastern Catholic Churche”s, Zenit, 3 November 2011. http://www.zenit.org/article-33766?l=english
2. Apparently the theological adjustment was gradual:
The first theological principle to be accepted was the general Latin ecclesiology, that is theology of the Church and therefore automatically the Catholic theology of the papacy. Secondly other theological doctrines and opinions such as the Filioque and the conception of purgatory were accepted. Later the theological definitions of the Immaculate Conception, the Assumption of the Blessed Virgin and the Vatican I definition of papal infallibility and jurisdiction were to be accepted.
“Catholicism in the Ukraine: Unresolved problems”, Church Unity. http://www.catholic-church.org/church-unity/gk_cat_e.htm
3. Roman Woronowycz, “The Pontiff’s Ukrainian roots”, The Ukrainian Weekly, Vol. LXIX, No. 26, p. 17, 1 July 2001. http://www.scribd.com/doc/12842698/The-Ukrainian-Weekly-200126
4. “Re: The Pontiff's Ukrainian roots”, The Byzantine Forum, Hritzko, 15 September 2004. http://www.byzcath.org/forums/ubbthreads.php/topics/3506/Re:%20The%20Pontiff's%20Ukrainian%20ro
5. “Joint International Commission for the Theological Dialogue between the Roman Catholic Church and the Orthodox Church: Seventh plenary session”, Balamand school of theology (Lebanon), 17-24 June 1993.
6. Judith Ingram, “Vatican, Orthodox Hierarchy Talk in Moscow”, AP, 17 February 2004.
7. Archbishop Thomas Gullicksonon 23 September 2014, quoted in “Russian expansion endangers Catholics in Ukraine, nuncio warns”, CNA, 29 Spetmber 2014. http://www.catholicnewsagency.com/news/russian-expansion-endangers-catholics-in-ukraine-nuncio-warns-86969/
8. “Slavonic Language and Liturgy”, The Catholic Encyclopedia, 1912. http://www.newadvent.org/cathen/14041b.htm