The difference, such that there is one, is rather simple but I suppose also somewhat difficult to grasp if you assume the two to be
opposing sides (which I suppose they have turned out to be historically since Chalcedon, but I have no need for Chalcedon, so I don't think it's necessary to battle it to describe how we see things; the difference existed before Chalcedon), rather than a matter of differing emphases born out of preexisting differences between the Antiochian (dyophysite) party and the Alexandrian (miaphysite). Quite simply, we do not speak of Christ as being
in two natures after the union (the incarnation), while the dyophsites do.
A selection of relevant portions from the Coptic liturgies and hymns, to illustrate the point:
"Amen. Amen. Amen. I believe, I believe, I believe and confess to the last breath, that this is the life-giving body that your only-begotten Son, our Lord, God and Saviour Jesus Christ took from our lady, the lady of us all, the holy Theotokos Saint Mary.
He made it one with his divinity without mingling, without confusion and without alteration. He witnessed the good confession before Pontius Pilate. He gave it up for us upon the holy wood of the cross, of his own will, for us all. Truly I believe that his divinity parted not from his humanity for a single moment nor a twinkling of an eye. Given for us for salvation, remission of sins and eternal life to those who partake of him. I believe, I believe, I believe that this is so in truth. Amen."
-- The priest's confession before the Eucharist, Liturgy according to St. Basil
"Hail to the uniting place of the unparted natures, that came together in one place, without ever mingling."
-- from the Wednesday Theotokia
"For He who was born is God, born without pain from the Father, and He was also born according to the flesh, without pain for the virgin.
One nature out of two, Divinity and Humanity, wherefore the Magi silently worship, uttering His Divinity."*
-- from the Gospel response for the Nativity
"The ark overlaid, roundabout with gold, that was made with wood that would not decay. It foretold the sign of God the Word, who became man without separation. One nature out of two,
a holy divinity, co-essential with the Father, and incorruptible.
A holy humanity, begotten without seed,
co-essential with us, according to the Economy."
-- from the Sunday Theotokia of the midnight praises (Tasbeha)
"Hail to the bridal chamber that is adorned with all types, for the true Bridegroom Who became united with humanity."
-- from the Verse of the Cymbals for the crowning ceremony
"From you the Divine appeared, Christ the King of glory united with Humanity"
-- from the praises before the Monday Theotokia during the month of Kiahk
"O people who love Christ, come see and be amazed about this mystery that was revealed to us today. For our Lord Jesus Christ, gathered with His virgin mother and our fathers the apostles, He revealed to them His divinity.
Six jars of water changed into chosen wine, through His great glory, in the Wedding of Cana of Galilee.
He who sits on the Cherubim revealed His divinity, He performed signs and wonders, and sat with men as God.The coessential of the Father, who is before all ages, today was in the Wedding of Cana of Galilee."
-- The doxology for the Wedding of Cana
etc., etc., etc.
All of these are available for your perusal at tasbeha.org's hymns library.
As you can see, we have no problems whatsoever fully affirming and proclaiming that Christ our God is 100% wholly and completely divine, and 100% wholly and completely human, and that there is neither a separation of the two natures (and yes, we can and do talk about them in that way; recall that we praise the Theotokos as "the meeting place of the unparted nature
s"), nor a mixture of the two, nor did either one at any point consume or overwhelm or dissolve the other.
The one point of difference, or difference in emphasis, that you are likely to find is that we do not talk about Christ being
in two natures after the union, following our understanding of HH St. Cyril that we are not to divide the natures after the union, but to proclaim one Son -- one incarnate nature of the Word.
And as far as I can tell, we don't have any problem with St. Cyril's idea that we can admit
in theoria, in contemplation that there are two, insofar as it's not like either loses its distinctiveness with the union, since it's not a mixture or confusion -- what is human about Christ in terms of His actions (e.g., that He ate, and slept, and wept, etc.) is indeed befitting His humanity, which He did indeed truly embody (i.e., it did not just
appear to be so, as the Gnostics would have it -- it was true flesh, same as yours or mine), just as His divinity is testified to by His miracles, as we proclaim about the miracle of the wine at the wedding of Cana above. Indeed, if we had such a problem with that, we couldn't have these hymns and responses and prayers that we clearly do have. Christ not Christ if He is somehow not both, as that is the entire point of the incarnation in the first place.
Just the same, in terms of how we will talk about the incarnation as an event, or rather the 'result' of it or 'what it means' (sorry for all the scare quotes; truthfully, I find this just as annoying to talk about as you find it to research, as it's like splitting a gnat's hair 1,600 different ways), we simply stick with "One incarnate nature of the Word", which is perfectly Orthodox, and to us has some advantages over the dyophysite understanding as enshrined in the Tome. As our father HH St. Severus of Antioch put it (rhetorically!), how can one divide walking upon the water, when to walk upon water is alien to man, while to use feet is alien to God? The point being that
the action is belonging to
one and the same Person -- the God-man, Jesus Christ. Hence we do not have this "This nature does/receives this, while this nature does/receives that" dichotomy that the Chalcedonians have from having accepted the Tome, with which our major point of contention was and is precisely those portions which say "this nature receives insults, that nature receives glory" (or some such; I don't carry it around with me every day

).
Hopefully this helps at least a little bit.
. . . . .
* - Boring linguistic stuff (feel free to skip this if you didn't notice how weirdly phrased the starred bit is, or if you don't care): Not that it affects the point I'm trying to make one bit, but since it bothers me to have to put it out there when as a native English speaker it makes very little sense to me and I can't explain it, I have reason to wonder whether the confusingly-worded final clause has been mistranslated from the Coptic into Arabic, and from there into English. I don't
really know Coptic, but I can read it, and the Coptic ends
khen fai seoousht emmof nje nimagos, ewkho enroou ewertheologin. From what I can piece together from a very general level of background knowledge and being able to use Crum's Coptic dictionary, the first part is pretty unamibiguously 'therefore/in that way [khen fai] worship [seoousht] Him [emmof; dative -- "to him"] the Magi [nimagos; nje is just the nominal subject marker which proceeds nimagos, since the Magi are the subject], kho enrou (be silent/silently) ewertheologin (??) - I'm assuming this is some kind of masculine possessive third person form of "divinity", since the root noun is clearly from the Greek, but I don't see how they're getting "
uttering His divinity" from it. To utter, cry, announce, etc. in Coptic is ōsh, esh-, sh-, osh (depending; since Coptic loves making large compound words, basically every root noun has multiple forms that change depending on what kind of construction it is being used in), none of which are found attached to the root in that word. It's highly possible/probable that the entire thing is Greek and I just don't know it, since all the other words I can find at a glance in the dictionary that begin with ev/ew appear to be Greek (evlogite, evxenos, ewfimia, Ewfratis, etc.), but I don't have a Greek dictionary, so that doesn't help. At any rate, this has been translated into Arabic as سجد له المجوس. ساكتين وناطقين بلاهوته 'the Magi prostrate before Him, silently speaking His divinity'. Hmmm...