What is the difference between a Christian and a Messianic Jew?

Messianic Jews are Jewish and acknowledge that Yeshua is their Messiah. There are Christians that keep the Jewish traditions and may aspire toward worshiping G-d in that way but they are not recognized as being Jewish in an authentic Messianic congregation. One cannot attend an ethnic Church and now say they are of that ethnicity just because they feel at home in their worship. One is not Jewish because they attend a Messianic Congregation.
Shalom, E

A Messianic Jew is simply a modern name for a Jewish Christian who does not renounce Judaism. Some are virtually indistinguishable from other Christians, but the majority continue to affirm some degree of Jewish identity. The Jews for Jesus movement characterizes themselves as "Ethnic Jews, embracing Jesus, still Jewish." Within that broad category there are two basic types: those that keep Jewish customs and those that do not. There are also further distinctions in terms of how "observant" Messianic Jews follow the Jewish dietary and purity laws. This follows the pattern among religious Jews generally.

A significant difference between most Messianic Jews and "normal" Christians is that Messianic Jews generally practice circumcision as a sign of God's continuing covenant. Also, while they affirm that Jesus ushered in the New Covenant, and they hold the New Testament to be holy scripture, they generally reject supersessionism, a doctrine taught by many Christians which can be interpreted to mean that God's covenant with the Jews has ended. Also:

• They tend to call their pastors "Rabbi" instead of "Reverend" or "Father."
• They celebrate Jewish holidays.
• Services may include traditional Jewish prayers, especially the Shema, which Jesus is recorded a reciting verbatim in the Gospels (Mark 12:29).
• They observe the Sabbath on Saturday, and some follow Jewish Sabbath laws.
• Some [not the majority] Messianic Jews do not accept the Christian doctrine of the Trinity
• Although virtually all practice Baptism [usually full immersion] they do not practice all Seven Sacraments as Catholicism and Orthodoxy do.

This site offers a survey of Messianic Jewish traditions. It appears to be mostly accurate. One exception is its claim that Messianic Jewish services do not include Holy Communion. That may be the case for some but definitely not all such traditions. It is correct when it says:

A Messianic congregation can be a very diverse group, including Jews who carefully follow Jewish laws, Jews who have a more liberal lifestyle, and individuals who do not follow Jewish laws or customs at all.

See also: Messianic Jews: A Brief History

Incidentally, contrary to the OP, outside of Israel, Messianic Jews do not necessarily speak Hebrew, depending on how much, if any of it, they inherited growing up.

I was born and raised an orthodox Israeli Jew, and was an orthodox Rabbi for many years before becoming Messianic in 2008. The main body of beliefs are the same for Christians and Messianic Jews. However, the way we interpret scripture, and practice our faith can be very different indeed. Shabbot shalom to you all.

The term Messianic Jew refers to Christians who identify as Jewish, as compared to those who identify as Russian, English, Thai etc. Most of them are of Jewish descent, though there are some who are not but have chosen to identify themselves as Jewish.

Most Messianic Jews believe there is a very strong continuity between the Hebrew scriptures and Christianity. Many uphold the Mosaic Law and, for example, celebrate its festivals. They believe that Jesus is the fulfilment of many of the prophesies of the Hebrew scriptures, just as Christians do. In general their beliefs are likely to be largely the same as other Christians, though there's also just as much diversity among them as there is the rest of us.

There are also Christians who are of Jewish descent but who do call themselves Messianic Jews. Jewish identity is more complex than other ethnicities as it may involve an identification with the Jewish people, their history, their culture and the religion of Judaism, or it may involve an identification with only some of those things. These people may acknowledge their Jewish descent but not strongly identify with the Jewish culture or religion, perhaps because contemporary Judaism has significant differences from how it was practiced in Biblical times.