The phrases 'Jewish Christians' and 'Christian Jews' sound like a contradiction in terms, but there are such people to whom these phrases can apply. Jewish Christians and Christian Jews mean the same thing; they are also known as Hebrew Christians.
Meaning of the words
In addition to having several synonyms, the words Jewish Christians can mean two things depending on whether the reference is a historical or a present day one. In a historical context, Jewish Christians are the early Christians, who belonged to the Jewish faith, but were drawn towards the teachings of Christ and the philosophy of Christianity.
Here, the terms Jewish Christians and Christian Jews are used with reference to the interconnectedness of the traditional Jewish beliefs and customs, and the tenets of the newly developing (as it was at that time) Christian faith. When early Christianity is being discussed, you will often hear the terms Jewish Christians and Christian Jews.
Judaizers
Some of the first Jewish Christians thought that those who were not Jews should embrace Judaism and follow a Jewish way of life. These Christian Jews were called Judaizers, and were looked down upon. Interestingly, the apostle Paul is believed to have referred to Jesus' disciple Peter as one of the Jewish Christians in an insulting manner in front of several others.
Modern Sense
When the phrases Jewish Christians, Hebrew Christians and Christian Jews are used in a modern sense, they refer to people who have a Jewish heritage, culture or ancestry. However, such people follow Christian tenets and philosophy, rather than Judaism. The terms Jewish Christians and Christian Jews are also applied to those who have formally converted to the Christian faith, and to those who do not practice the Hebrew faith, due to the circumstances of their upbringing, for example. The appellation Christian Jews can be used for those who are Jewish, but have been absorbed into the Christian majority. Although Christian Jews and Messianic Jews both represent a blend of Judaism and Christianity, Christian Jews are quite different from Messianic Jews, who are definitely Jewish, and only differ from other Jews in that they believe that Jesus was the true Messiah.
Christian Jews
Today, there are some groups of people, whose culture indicates both Jewish and early Christian influences - they can be called Christian Jews, too. These include the Syrian Christians of India (whose Jewish heritage has largely been lost), the Knananites (a sub sect of the Syrian Christians), who are descended from early Jewish Christians who came to Kerala in India in 643 AD, and the Fallasha of Ethiopia, who have a culture similar to that of Christian Jews. The Syrian Christians, who are also called the Nasrani (from 'Nazarene'), the Syrian Malabar Nasrani and the St. Thomas Christians, are aware of their Jewish origins, but follow the Christian faith. Their ancestors were converted to Christianity by St. Thomas, who came to India as a missionary. The Knananites follow Oriental Orthodox and catholic streams of Christianity, but they are strictly endogamous. The Ethiopian community, the Fallasha reflects an interesting Jewish tradition even today.