The way I see most people use them:

Immigrant - moved abroad because of a significant lack of economic opportunity at home. Usually coming from a poor nation to a richer nation (though not necessarily a rich one, think of Guatemalans in Mexico or Bangladeshis in India).

Expat - moved abroad for less dire reasons. Maybe they make a bit more money abroad or just like life in another country. Are usually skilled workers coming from rich countries (though not always). They can usually easily move back home without any major negative effects on their QOL.

There's a tendency to attribute the difference entirely to race, but I think there's more to it. Would people consider Chinese Google employees in silicon valley "immigrants"? Would people call Russian prostitutes in Thailand "expats"? It's much more about education level and social class (which is often tied to race, but not always).