In both cases the person is leaving the country, he was born in, and he is moving to another country on a permanent basis. He plans to settle down there.
The word `emigrate’ is normally used to refer to the movement from a place. When the person leaves his country of origin, he is emigrating.
For example, when somebody is leaving India and moving to New Zealand, I can say that he is emigrating from India to New Zealand.
The person who `immigrates’, moves to another country. In this case, you are talking about the movement in terms of the country the person is going to settle in.
Example: During the Second World War, lots of Germans immigrated to the United States.