The significance of this case is that it established the idea that the 5th Amendment protects naturalized citizens from being discriminated against because they are not native-born citizens.
At this time, there was a law that said that a naturalized citizen who went back to his or her native country and stayed there for more than three years would lose his or her American citizenship. There was no provision in law that said that a native-born citizen could lose his or her citizenship for living abroad for any length of time.
Angelika Schneider was a naturalized American citizen who lost her citizenship because of the provisions of this law. The Court held that it was illegal for the government to do this to Schneider. It held that this law was discriminatory because it treated different types of citizens differently. This, the Court, held, was illegal under the Due Process Clause of the 5th Amendment.
Thus, this case is significant because it protects the rights of naturalized citizens. It establishes the idea that naturalized citizens cannot be treated differently than people who were born in the US and have been citizens since they were born.