Both mean *to come* to another country to settle; however, emigrate also means *to leave* one country to settle into another.
That is what my Websters says, anyway. I certainly do not see how a person can *come* to a country to settle in, without *leaving* his previous country, so it would seem that immigrate also means *to leave*

Wow, this is a very good question! According to my Websters, it is not the same word with different spellings, because the two words are listed under their respective letters, instead of under one letter, with a variant spelling.

I am sure that some of the more knowledgeable posters here can shed some better light on this, than can I!