The Ebionites and Nazarenes were Jewish–Christian sects during the Second Temple era. We know that the Ebionites believed Yeshu was born of a natural generation between Joseph and Mary; while some Nazarenes believed in a virgin birth. How were the two different from orthodox (Pharisee) Judaism? For example, did both sects pray "Our Father" in Aramaic or in Hebrew?
According to some notes provided by Eastern Orthodox monk Seraphim Rose to the English translation of Protopresbyter Michael Pomazanski's Orthodox Dogmatic Theology, the Ebionites and Nazarites were both groups of heretical Judaizers.
The Ebionites, he wrote, "considered Jesus Christ to be a prophet like Moses; they demanded of all Christians the strict fulfillment of the law of Moses; they looked on the Christian teaching as a supplement to the law of Moses."
The Nazarites "believed in the Divinity of Jesus Christ, but insisted on the fulfillment of the law of Moses by Christians who were Jews, without demanding this of the non-Jewish Christians (moderate Ebionites)."
[I noticed that you use the term "Nazarene". I assume "Nazarite" refers to the same group.]
In addition, there was another group of Ebionites that Fr. Seraphim refers to as "Ebionite-Gnostics", whose teaching was composed of the teaching of the Jewish sect of the Essenes, who lived on the Dead Sea (Qumran excavations, the “Dead Sea Scrolls”), joined to the elements of Christianity and Gnosticism." "The Essenes," he further wrote,
considered themselves the preservers of the pure religion revealed to Adam but later obscured in Judaism. The Ebionite-Gnostics recognized the restoration of this religion by Christ, as the bearer of the Divine Spirit; the Gnostic element was expressed in their view on matter as being an evil principle, and in the preaching of severe asceticism.
Irenaeus of Lyon wrote of the Ebionites in the 2nd century:
Those who are called Ebionites agree that the world was made by God; but their opinions with respect to the Lord are similar to those of Cerinthus and Carpocrates. They use the Gospel according to Matthew only, and repudiate the Apostle Paul, maintaining that he was an apostate from the law. As to the prophetical writings, they endeavour to expound them in a somewhat singular manner: they practise circumcision, persevere in the observance of those customs which are enjoined by the law, and are so Judaic in their style of life, that they even adore Jerusalem as if it were the house of God (Against Heresies, I.XXVI.2)
Eusebius of Caesarea also wrote of the Ebionites and the Nazarenes in his 4th century Church History:
The evil demon, however, being unable to tear certain others from their allegiance to the Christ of God, yet found them susceptible in a different direction, and so brought them over to his own purposes. The ancients quite properly called these men Ebionites, because they held poor and mean opinions concerning Christ.
For they considered him a plain and common man, who was justified only because of his superior virtue, and who was the fruit of the intercourse of a man with Mary. In their opinion the observance of the ceremonial law was altogether necessary, on the ground that they could not be saved by faith in Christ alone and by a corresponding life.
There were others, however, besides them, that were of the same name, but avoided the strange and absurd beliefs of the former, and did not deny that the Lord was born of a virgin and of the Holy Spirit. But nevertheless, inasmuch as they also refused to acknowledge that he pre-existed, being God, Word, and Wisdom, they turned aside into the impiety of the former, especially when they, like them, endeavored to observe strictly the bodily worship of the law [Eusebius seems to be referring to the Nazarenes here].
These men, moreover, thought that it was necessary to reject all the epistles of the apostle, whom they called an apostate from the law; and they used only the so-called Gospel according to the Hebrews and made small account of the rest [Scholars disagree whether Eusebius is referring here to the Gospel according to Matthew, as did Irenaeus, or some other spurious Gospel].
The Sabbath and the rest of the discipline of the Jews they observed just like them, but at the same time, like us, they celebrated the Lord’s days as a memorial of the resurrection of the Saviour.
Wherefore, in consequence of such a course they received the name of Ebionites, which signified the poverty of their understanding. For this is the name by which a poor man is called among the Hebrews.
On the basis of the patristic sources in our possession, I would say that a big difference between Ebionites and Nazarenes was their respective opinion towards Paul of Tarsus: while in fact all the ancient writers affirm that the Ebionites despised him, calling him "apostate of the Law", according to what Jerome reports in his commentary on Isaiah, the Nazarenes had a high opinion of Paul, considering him the one thanks to whom true faith reached the Gentiles.
Jerome's comment to Isaiah 9: 1 (8:23 in the Masoretic text)
"In the former time he debased the land of Zebulon, and the land of Naphthali; But in the latter time he hath made it glorious: Even the way of the sea, beyond Jordan, Galilee of the nations "
"The Nazarenes, whose opinion I have reported below, try to clear up this passage in this way: with the coming of Christ and the shining of His preaching, the land of Zebulon and the land of Naphthali was freed from the errors of the scribes and Pharisees , and it shook the very heavy yoke of Jewish traditions from their neck. Later, however, the preaching increased in intensity, thanks to the Gospel announcement of the apostle Paul, who was the last of all the Apostles; and the Gospel of Christ shone up to the borders of the Gentiles and the ways of the whole sea. Finally the whole world, which previously walked or sat in darkness and was wrapped in the chains of idolatry and death, was able to contemplate the clear light of the Gospel "