The relationship is truly more strained between Moscow and Rome than, let’s say, Constantinople and Rome, Bucharest and Rome, or Athens and Rome. The Patriarch of Constantinople and the Pope met each other many times since the mutual anathemas were lifted by EP of Constantinople Athenagoras and Pope Paul VI. Also, Pope John Paul II visited Romania and met Romanian Patriarch Teoctist in 1999 (there might have been visits by the Romanian Patriarch to Rome as well). And Pope John Paul II visited Athens and met the leader of the Church of Greece. But no Russian EO Patriarch would visit Rome, and the Pope is not welcome in Moscow. Pope John Paul II never realized his fervent dream and desire of visiting Moscow and Russia, because Moscow made it clear that he wasn’t welcome to visit.
I will add a few observations of my own, partly derived from my involvement with Russian Orthodox faithful and clergy (clergy of ROCOR - Russian Orthodox Church Outside of Russia), partly from surfing the internet, especially Russian EO websites affiliated with ROCOR, and Interfax Religion (Interfax is a Russian news service) and occasionally Ukrainian news sites.
One of my first experiences was that the Calendar is huge for Russian EO folks. Yes they consider us Catholics as heretics on account of us using the New Calendar (Gregorian Calendar instituted by Pope Gregory in the 16th century), but they also consider the EP of Constantinople, the Greek EOC, Romanian EOC, and so on, as heretics for the same reason. When Rome invented the New Calendar (Gregorian Calendar), the various EO Churches held many pan-Orthodox conferences and local conferences and issued a string of condemnations and anathemas against the New Calendar, starting with the late 16th century and up until the early 20th century. Then, around 1922, the EP of Constantinople transitioned from the Old Calendar (Julian Calendar) to the New Calendar (Gregorian Calendar), and his move was followed by some but not all EO Churches. Well, the Russians not only didn’t transition to the New Calendar, but regard the EP and others as heretics since they reversed a position they held since the late 16th century, namely a position of condemnation of the New Calendar. The ROCOR priest whose church I visited many times, told his parishioners to stay away from New Calendar churches such as the Greek and Antiochian EOC, whenever possible. So, my point is, the Russians are not exactly in love with us Catholics, but they also condemn, quite harshly, the New Calendar EO Churches.
The adoption of the New Calendar by the EP was seen, rightly so, as a concession to the spirit of ecumenism and reconciliation with Rome, and ecumenism is another heresy and big NO-NO as far as the Russians are concerned.
If the adoption of the New Calendar was bad, the mutual lifting of the 1000-year old anathemas in 1965, by EP Athenagoras I and Pope Paul VI, was seen as more than bad, it was seen as an even worse heresy. Metropolitan Philaret of ROCOR wrote a series of Sorrowful Epistles (open letters) addressed to the EP, protesting the lifting of the anathemas. Also, the Holy Synod of ROCOR anathematized specifically the ecumenical movement later on during the 1970s-1980s. See:
orthodoxinfo.com/ecumenism/philaret_lifting.aspx
romanitas.ru/eng/The%20Anathema-Lifter%20Fr.%20John%20Shaw.htm
As an aside, I recently stumbled upon a biography of the late Metropolitan Philaret of New York of ROCOR, describing his great courage in the face of Soviet and Chinese Communist persecution, and his miraculous escape to freedom in America. I was totally taken aback - the man was a very courageous, and very holy person. He is now regarded as Saint Philaret and Confessor by the Russians, and I would like to venerate him as such, even though I obviously think that he was in error when he rejected the authority of the Pope.
I hope I demonstrated, somewhat, the Russian standpoint as I observed it up close and personal among my Russian acquaintances. And I haven’t even mentioned yet that they also condemn the use of musical instruments (as opposed to human voice only) during Divine Liturgy, and the presence of pews in such churches as the Greek and Antiochian. So, do they condemn us Catholics? Yes, on account of many things, especially the New Calendar, but also the musical instruments, pews, leavened bread, no beard on clergy, and so on. But my point is, they condemn just as much (or almost as much) the Greek, Antiochian, Romanian New Calendar EO Churches, and the ROCOR priests advise the faithful not to attend those churches except for reasons of economy when the Russian Church is not available, and also the ROCOR priests advise the Russian faithful not to present themselves for Holy Communion in the New Calendar EO Churches (except, again, in cases where an Old Calendar church and priest is not available to administer the Sacraments).
This great emphasis on details might have something to do with the Russian psyche. Let’s not forget, there were thousands and thousands of Russian Old Believer martyrs, who chose imprisonment, torture, and death, over issues that seem even more minor than the Calendar, or the presence of pews and musical instruments in churches. Essentially, when the leaders of the Russian EOC and the state imposed some minor modifications (minor to us, outsiders, but not to them!!!) to the way of making the sign of the cross, and to liturgical prayers, that lead to a significant schism within the Russian EOC, where the state imprisoned, tortured, and executed many Russian “Old Believers”, who refused, for example, to make the sign of the cross using three fingers joined instead of the traditional way of two extended fingers only. It may seem minor to us, but people went to prison and died over this issue.