Both the Assyrian Church of the East and the Ancient Church of the East, which you are referring to, have a larger presence in Iraq than Iran (their largest concentration of members being in the Nineveh Plains), however, it is true the Assyrians are one of the two main Christian denominations in Iran, the other being the Armenian Apostolic Church. In Iraq, the other traditional denomination is the Syriac Orthodox Church, and then in the area around Baghdad you have the Chaldean Catholic Church, which broke away rrom the Church of the East in the 17th century. Today, most Chaldeans speak only Arabic, whereas of the million or so Assyrians, at least 700,000 continue to speak a dialect of Aramaic (which is related to but not directly descended from Classical Syriac), making them the largest Aramaic speaking population in the world. Additionally, like in the Syriac Orthodox and Chaldean churches, the Church of the East uses Classical Syriac as its liturgical language, and uses the Peshitta as its Bible.
The Assyrian Church of the East and the Ancient Church of the East became separated as a result of a schism in the late 1960s after some bishops of the Church of the East discovered their hereditary Patriarchate was a violation of their own canon law, and fuel was added to the fire when the Assyrian Church of the East adopted the Gregorian Calendar, so for a time, even after the assasination of the last hereditary Patriarch and the election of the reforming Patriarch, Catholicos Mar Dinkha IV, memory eternal, the Ancient Church of the East under its own Patriarch Mar Addai II, memory eternal (who was just 20 when he was consecrated as Patriarch) existed as a sort of Old Calendarist alternative to the much larger Assyrian Church of the East, with about 100,000 members to the million or so in the Assyrian church. Additionally, in the 1920s, the Patriarchate of the Assyrian Church of the East had been relocated to Chicago (which is home to a huge Assyrian diaspora), but the Ancient Church of the East established its rival Patriarchate firmly on the shores of Iraq. Recently, since the death of Mar Dinkha IV, the Patriarchate of the Assyrian Church of the East relocated to Iraq, and the two churches have been in the process of reunification for some time now, so hopefully within the next decade or so there will once more be a single unified Church of the East, as was the case since the first century AD.
The Church of the East does have a particularly glorious past, for at its peak, it was the largest church in the world in terms of geographic territory, extending from Aleppo to Mongolia and from Socotra (an island off the south coast of Yemen) to Tibet, but then in the 13th century, the evil Muslim warlord Tamerlane initiated a genocide against the Assyrian church and the large number of converts of different ethnicities who were members of it, which resulted in its members being killed off everywhere except India and the Fertile Crescent. I am at a loss as to why Tamerlane is celebrated as a national hero in Uzbekistan; to my eyes the tomb of Tamerlane in Tashkent produces a similar emotional reaction to what I expect I would feel if Germany built an elaborate monument to Adolf Hitler in the center of Berlin.
At any rate, the Assyrians are absolutely lovely people and there are a large number of Assyrian churches in the United States and Europe, both of the Assyrian Church of the East and the Ancient Church of the East.
Additionally, unlike the Eastern and Oriental Orthodox, the Assyrians allow anyone who believes in the real presence of Christ in the Eucharist to take communion in their churches. So I would strongly encourage anyone interested to visit the nearest parish of the Church of the East. They are extremely friendly and loving people.