I think "the sharing of the same table" was perhaps the hardest thing I've ever had to understand, as unfortunately it is usually seen as a rejection. RE: Holy Communion; more than anything, when Orthodox share the same table, they are accepting all that they teach as true, of course. To tell the truth, this is something which made absolutely no sense to me a little while ago, as it initially looks prideful to take this course. But, in reality - counter-intuitively - it is quite the opposite. This is my understanding: to share communion with another Church would be to accept all that they teach as true; either that or we affirm that the difference in our beliefs are unimportant, which is to say the beliefs themselves are unimportant. This is to show arrogance towards the beliefs of the other Church and disdain towards our own. I know people who have converted from Anglicism and Catholicism to Orthodoxy. Most of them had some stage where they continued to attend their old Church, but for reasons of conscience, did not take communion. No priest told them to do this. They simply recognized that what their old Churches proclaimed was not what they believed. They knew in themselves that to take communion in their old Churches would be to lie to their fellow Anglicans/Catholics and themselves. I help out at my local Methodist Church (that is, I help the children in Sunday School) but I don't receive communion in my Methodist Church. I also attend an Orthodox Church and I have discovered, being in an Orthodox Church with the sights, sounds, smells and actions is simply another level to reading the beautiful prayers, etc. My Orthodox friends, obviously, have family members and friends who are protestants; they love their friends dearly, of course, and they pray with them during grace before meals, in their home chapel, etc.
My protestant friends say. ."Jesus said "Do this in remembrance of Me" When I share communion I am remembering Jesus death. The theology of the fellowship is largely irrelevant". Also, they don't understand why Orthodox and Catholics can't "overcome their differences" I completely understand where they are coming from, as I used to feel the same. My usual answer is:
First of all, there is a difference in beliefs on the nature of the Sacraments. On the individual Sacraments, there are also differences. First, re the Catholics, the Orthodox and the Eucharist: In the Orthodox church, the culmination of the service is the epiklisis which is when the Holy Spirit is called down to act in the Eucharist:
"Send down Thy Holy Spirit upon us and upon the Gifts here spread forth, and make this bread to be the precious Body of Thy Christ... ."
This prayer is not used in Catholic churches, and Catholics believe that when the priest says "This is my Body..." "This is my blood..." that the change occurs. This shows how in the West the Body and Blood of Christ is believed to become present by the act of the priest, rather than through the prayers of the entire church. Also, in the Orthodox church the body is placed in the chalice, and so both body and blood together are given to the Orthodox using a long spoon. And what of the bread used? The normative practice in Catholic churches is to use wafers, hence all the disparaging comments about "crackers" from Protestants and atheists alike. In the Orthodox church we always use leavened bread, because by using unleavened bread a key symbol of Christ being the new leaven is completely absent. These acts are sacramental acts - not magic - so the issues I have are not based on "such and such needs to happen otherwise the change doesn't occur". They are based on the fact that such acts show forth a truth and teaching of the Church. Change the act - alter the symbolic elements used - and you change what you are proclaiming.
And what is this practice of Eucharistic adoration? The gifts at Holy Communion are for the faithful to share, i.e. you eat them. The act of "saving" these items and "exposing" them for adoration is completely alien to the Orthodox understanding on what the gifts are and what the Eucharist is for (i.e. it's not for the purposes of "making" Christ's flesh so that it can be stored away and worshipped at various times). There are of course many other ways in which the Liturgy of the Orthodox differs from the Mass of the Catholics: that the priest should normally face East, the fasting canons for preparation, the psalms and Bible readings used, and of course the Creed that is said during the Liturgy has a very real difference.
There are differences too in the other Sacraments. The Catholics, and most Reformed faiths too, do not believe that humans can have real communion with God through His uncreated divine energies toward creatures, from the Father through the Son in the Holy Spirit. Instead, the teaching is that God "creates" Grace through which we mere mortals may experience the unknowable God. The Catholic teaching is that Heaven is the unmediated vision of the Holy Trinity. The Orthodox believe that such a vision can be attained by some saints even while still on earth, and that in Heaven we will experience the Divine in every way - not just as "a vision". These are fundamental differences in Who God is and how we know Him. I will not even begin to go into the differences regarding the Theotokos, purgatory, substitutional atonement, and of course, the view of the Pope's authority.
And these differences in belief all effect our view of Who God is. These differences also directly influence the practices of each church, so that the belief leads to the act, and the act reinforces the belief.
I remember some time ago I talked about there being only one human nature. I said at the time, that the power of human empathy cannot be underestimated. We are all united through a common humanity. Some of us will feel more united through the love of the same things. A smaller group of us will feel united specifically through a shared love of truth. Within that group, we will see more unity in those with whom we can actually agree what that truth is. These unions should not be despised. But the unity of the Church is sacramental and holistic, not depending upon our shared ideas and opinions, but of shared beliefs. There are differences within the Orthodox churches, and amongst the Orthodox churches, between Orthodox families and within Orthodox families. Differences between the Orthodox and non-Orthodox only cause strife when people insist they must be the same.