Theologically, none. Different Orthodox Churches have their own cultural and historical markers, language, slight variations of vestments, even external praxis and non-canonical parts of the Divine Liturgy (for example, the Greeks don't use the Beatitudes in the Third Antiphon).
The reason why all Orthodox can commune in any Orthodox church, regardless which Patriarchy they are from, is because all particular Churches that form the Orthodox Church community profess the same faith. Communion is the expression of, and not a means of establishing, theological union.
The change occurred in 1923 under the Ecumenical Patriarch Meletius (Metaxakis), a satanic character, heretic, and a Freemason who on his own implemented the "Revised Julian Calendar" which will for the next few centuries correspond to a very close proximity to the Gregorian calendar, but eventually will diverge from it.
His decision to change was not approved by an all-Orthodox Synod, so the decision was un-canonical. The reason for the change is also suspect. The "Old Calendar" served the Church for 2,000 years quite well, so no particular urgency existed to drop it. However, the heretic EP Meletios (who has yet to be condemned publicly by his successors!) did this as part of his satanic drive to "westernize" the Orthodox Church. He armwrestled those churches that belong to him (the Church of Greece, GOARCH, etc.) to accept his changes; many others, for various political and other reasons followed suit for. hese canges, in addition to the Calendar included clean-shaven Orthodox clergy and Catholic-like or Anglican-like ("dog") collars, among other things.
He even unilaterally "recognized" Anglican ordinations and proclaimed Orthodox-Anglican "re-union." He created quite a stir in America, where the jurisdiction of the Orthodox was under the Russian Patriarch, taking advantage of the October Revolution in Russia, to wrestle control of American Orthodox, establishing the Greek Orthodox Archdiocese in America.
The heretic EP Meletios also grabbed control of the Finnish Orthodox Church (formerly under Russian Patriarch), allowing it to celebrate Easter according to the Gregorian Calendar (Easter is still celebrated in Orthodoxy by the Old Calendar by all EO Churches, save for the one in Finland).
He also wanted the exiled Russian Church, which took refuge in Serbia, where it was allowed by the Serbian Church to operate independently there, to be placed under EP's tutelage. The Serbian Church refused, thus saving ROCOR from absorption by a heretic EP. Meletios was an abomination, equal to Nestor, and other heretic Bishops of Constantinople, yet the EPs to this date will not anathematize him or his satanic changes. I was told by one monk of a Greek monastery that the current EP, Bartholomew I, promised personally back in 2003 that he would roll back the changes made by Meletios (I have no independent confirmation of this), including the Calendar, but there seems to be no indication this will happen.
A good rule of thumb is: never trust any Ecumenical Patriarch completely. There were more heretics among them than any other bishops of any other traditional Church (Antioch, Alexandria, etc.)
As for the Calendar, although the majority of Orthodox Churches (save for four or five out of 14) in the world use the New Calendar uncanonically established by a heretic EP Meletios in 1923, the vast majority (85%) of all Orthodox in the world use the Old canonical Calendar (Russians make up about 80% of the world Orthodox).
As a result of Meletios's demonic appearance, and the weakness of some Churches, the Orthodox world now celebrates Christmas and all other feasts (save for Easter) separately. Our Orthodox brethren belonging to the "New Calendar" Churches apparently prefer to celebrate with the hererodox.
No doubt, satan must be exceedingly glad. He rejoices at the disunity and savors knowing how much pain the New Calendar has caused.