Amended Hungary-Vatican financial concordat (2013): summary
After 16 years and a change of government the Vatican was able to renegotiate the Hungarian financial concordat to get even more favourable terms. Naturally, the public was unaware of what was going on.
On 21 October 2013 amendments were signed to the 1997 financial concordat with the Vatican. This regulates the "financing of public (social) service and religious activities of the Catholic Church in Hungary, as well as some additional financial issues".  The amendment of the concordat was justified in terms of the altered legal situation: "new Constitution, new religious laws, changed the structure of the public administration)." 
Conservative government remakes constitution and law on religions
The Vatican only picks advantageous legal changes. It doesn't renegotiate concordats when dictators are overthrown and democracy introduced — that apparently isn't enough of a legal change!  (The only exceptions to this are when the new democratic regime threatens to cancel the concordat, and renegotiaton amounts to getting half a loaf, rather than none. This happened in Austria when its fascist regime ended after WWII and in Spain when the dictator Franco died.) .
In Hungary, however, the decline of democracy has presented new opportunities. After the 2010 election, the ruling conservative party Fidesz offered a favourable political climate which allowed the Vatican to expand its privileges. It replaced the Hungarian Socialist Party (MSZP) that had negotiated the original concordat in 1997, and this gave the Vatican an opening.
The new constitution [...] accepts conservative Christian social doctrine as state policy, in a country where only 21% of the population attends any religious services at all. The foetus is protected from the moment of conception. Marriage is only legal if between a man and a woman. The constitution “recognize(s) the role of Christianity in preserving nationhood” and holds that “the family and the nation constitute the principal framework of our coexistence.” While these religious beliefs are hard-wired into the constitution, a new law on the status of religion cut the number of state-recognized churches to only fourteen, deregistering 348 other churches. 
In addition to the new constitution, an equally controversial law on religion was introduced in 2011. This pruned the list of religions eligible for the "church tax". Classifying the smaller denominations as ‘cults’ increased the incomes of the bigger Catholic and Lutheran churches. It led to a protests by international human rights groups.  These widely deplored developments formed the "significant changes in the Hungarian legal system" that served to justify the concordat amendments.
"The great public event the Hungarian public didn’t know about"
As so often happens, the concordat was renegotiated "out of sight of the public". There was no parliamentary scrutiny or discussion, no press announcement, and, due to the tight control over the press through the 2011 media law, there was little chance that it would be picked up there. So when the concordat was signed in a ceremony in parliament by the two representatives on 21 October 2013, "the public was left largely unaware of what exactly was being celebrated in the parliament". 
In the photo the head of the Hungarian Bishops' Conference, Erdő Péter, passes the pen to Hungary's Deputy Prime Minister, Zsolt Semjén, for him to sign the concordat. The Cardinal Archbishop's happy smile is well justified by the added concessions granted to the Church.
These include a larger intake from "church tax", a doubling of the subsidies to Catholic education and new state subsidies to train priests:
— Hungary has a ‘church tax’ where tax payers register with a church and the church gets a percentage of their tax according to the registered numbers for the country. Under the 1997 agreement, where fell below an agreed sum (as when too many people left the church) the state agreed to top up the church tax to make up the deficit. Under the amended concordat, the churches automatically get that agreed ‘minimum’ sum anyway plus all the church tax, as well.
— Under the 1997 agreement the state paid 50% of Catholic teacher’s salary and other expenses of Catholic school and colleges. Now it pays 100% and regards them as normal state education institutions, even though they’re still not bound to the state curriculum or oversight and are therefore free to create their own course content.
— Also, while Hungarian trainee priests live in Vatican-owned flats free of charge, the amended concordat obliges all other expenses of their training to be met by Hungarian state. So the taxpayers of Hungary are now paying the Vatican to train priests in the same way as they pay a medical school to train doctors.
Here is a summary of the additional privileges given to the Vatican in the 2013 amendments to the 1997 concordat  The Hungarian text of the amended concordat can be found in the draft bill from November 2013 to implement it. 
["On the "financing of public (social) service and religious activities of the Catholic Church in Hungary, as well as some additional financial issues"] 
[signed 21.10.2013, ratified 10.02.2014]
♦ The Hungarian government agrees to cover costs associated with teaching government-mandated “faith studies” by contributing enough to increase instructors’ salaries to the average for Hungarian teachers.
♦ Hungary agrees to provide the same amount of funding for the creation and utilization of “faith studies” teaching materials as it does for state-mandated “ethics studies” materials.
♦ And it agrees to provide the same subsidies to private Catholic dormitories and student housing given by Catholic higher-education as it does to state educational institutions.
♦ The number of students receiving government educational grants while attending Catholic schools, but not studying faith-related subjects, must reach at least 5% of the student body. The government and the Conferences of Hungarian Catholic Bishops will deliberate on this matter every four years.
♦ The annual state scholarships given to Catholic institutions of higher-education will be no less than HUF 7.5 billion.
♦ Students attending Catholic institutions are to receive the same scholarships, grants, and other financial aid, that are provided to students attending state-run schools.
♦ "The Hungarian state acknowledges Peter Pazmany Catholic University is an eminent university recognised by the Holy See"
♦ The state will provide scholarships for 2,500 Catholic students who are to receive the same amount of financial assistance as students studying education.
♦ Property belonging to the Catholic church in Hungary is to receive the same “cultural” subsidies granted to similar state-run property. This refers to the operation/maintenance of monuments, artistic treasures, museums, libraries, and archives.
♦ The government agrees to transfer free of charge to the Papal Hungarian Church ownership of property belonging to the Rome-Hungary Institute (a government organization in Rome).
♦ The state agrees to provide the Catholic church an amount equal to 1 percent of total revenues generated from Hungarian income taxes, regardless of whether people indicate their desire to donate to the Catholic church or not.
♦ The annuities and compensation given by the Hungarian government to the church will be adjusted to reflect the changes in the Consumer Price Index.
♦ The National Statistical Office’s reporting of government’s compensation to the Hungarian Catholic church in previous years (for properties not returned) will be adjusted to reflect any changes in the Consumer Price Index. Starting in 2011, supplemental compensation for such properties was HUF 5.4 billion.
1. Its Italian descriptiopn is "modifica dell'accordo del 20.06.1997: Accordo tra la Santa Sede e l’Ungheria sulla modifica dell’Accordo, firmato il 20 giugno 1997 sul finanziamento delle attività di servizio pubblico e di altre prettamente religiose ("della vita di fede") svolte in Ungheria dalla Chiesa Cattolica, e su alcune questioni di natura patrimoniale"
2. Magyarország Kormánya, T/ 12975, (This is a draft bill from November 2013 to implement the concordat amendments.) http://www.parlament.hu/irom39/12975/12975.pdf
3. Muriel Fraser, "The left gets a modus vivendi, the right a concordat", Concordat Watch. http://www.concordatwatch.eu/topic-31871.834
4. Kim Lane Scheppele, "Hungary’s Constitutional Revolution", New York Times, 2011-12-19 http://krugman.blogs.nytimes.com/2011/12/19/hungarys-constitutional-revolution/
5. "‘No’ to the new law on religions: Open letter to Prime Minister Viktor Orbán", Budapest Times, 2012-04-22. http://www.budapesttimes.hu/2012/04/22/no-to-the-new-law-on-religions/
6. Szobota Zoltán, "Nyilvános a vatikáni szerződés módosítása: Betonba öntött kiváltságok", (Changing the Vatican's contract with the state: Privileges cast in concrete), Hetek, 2013-11-15
Translation by Monika Hartill : "A Szentszék és Magyarország között megszületett megállapodás módosításának előkészületei a nyilvánosság kizárásával zajlottak, és az ünnepélyes októberi aláírással egy időben sem tudhatta meg a közvélemény, mit is ünnepelnek a két fél képviselői - Semjén Zsolt miniszterelnök-helyettes és Alberto Bottari de Castello apostoli nuncius - a Parlamentben."
7. "Benjamin Novak, “Hungary updates agreement with Vatican”, Budapest Beacon, 2013-12-09
http://budapestbeacon.com/featured-articles/hungary-updates-agreement-with-vatican/ This part is a translation of the summary in “Nyilvános a vatikáni szerződés módosítása”, Index, 2013-11-08 http://index.hu/belfold/2013/11/08/nyilvanos_a_vatikani_szerzodes_modositasa/