I might have been traumatized by US visa process, but in my mind "immigrant" has a very specific meaning: moving to a country with the intent of settling there permanently. And the US is very strict on that meaning: when I went to the US my work visa was explicitly non-immigrant, and calling myself an immigrant at the US embassy when applying for the visa, or while crossing the US border would have resulted in the denial of my visa, or the denial of my entry in the US. At least that's what the attorney drilled into my head...

Expat literally means "living outside of your homeland". So by definition an immigrant is an expat, but an expat is not necessarily an immigrant. Sometimes expat is used in a stricter sense (also called "true expat") to refer to people sent abroad by their employer on usually very very favorable packages (the "expat packages"): everything paid (rent in prime neighborhoods, private schools for kids, plane tickets to visit home, etc.), on top of an indemnity for living abroad, on top of the regular salary paid home.

In my case I was an expat in the United States (only went there to work, no intention to stay), but then I immigrated to Japan (now a permanent resident, no intention to leave). It would have been wrong to call me an immigrant in the US, but I am both an expat and an immigrant in Japan.