There are important similarities, yet differences, between “Hebrew 
," also known as "Jewish Christians," and “Messianic Jews.
Consider This:

“Hebrew Christians”
Hebrew Christians are Jewish people who identify themselves as Christians.
They are (mostly) members of Protestant and Catholic congregations.
Usually, they are not so strict about observing Kashrut 
(Jewish dietary laws), the Sabbath, or the Feasts.
They are generally assimilated culturally into the Christian mainstream. 
They continue to retain a strong sense of their Jewish identity. 
They have a strong desire to pass on their Jewish identity to their children. 

Messianic Jews:
Messianic Jews consider themselves as their primary identity to be "Jewish".
Their belief in Yeshua (Jesus) is to be the logical conclusion of their "Jewishness."
They try to structure their worship according to Jewish norms. 
They circumcise their sons.
They (mostly) abstain from non-kosher foods.
They observe the Sabbath, and the Feasts.
They do not use the label "Christian" to describe ourselves.
They also have a strong desire to pass on their Jewish identity and 
culture to their children.