The only difference between Christian Jews and Jews is Jews don't believe in Christ
Not the only difference ... but a major one. We also have substantially different concepts of the nature of sin and the after life. Belief in a divine entity other than G-d is polytheism (the awkward concept of The Trinity notwithstanding) and a Messianic 'Jew' would be considered a Jew in apostasy. Since Messianic Judaism is based in Protestantism many Messianic Jews aren't Jews at all (not being born to a Jewish mother) and hence take on the identity of 'Jews' perfidiously.
Yes the same exact belief as the Pharisees and Scribes had that Jesus rebuked.
The washing of hands were not so important as belief.
The main thing is belief in Jesus not traditions and sedition.