It’s not a British/American thing - any nationality can be referred to as an expat. It’s all a matter of what you’re trying to emphasize. The term “expat” implies being in a different country and feeling like a foreigner - using the term suggests that there is a degree of culture shock or not feeling like you fully fit in. Foreigners will often look for expat communities for support. That may be why you’re noticing it with British and American foreigners - you can be a French expat or a German expat or any other nationality, but if English isn’t your first language you’re less likely to know the term.

You’re also less likely to hear an American or British person refer to people who come to the U.S. or U.K. as “expats” - the term "expat"implies inclusivity with other people who came from the same place, while “immigrant” carries the implication of someone from a different culture that came here. As a native English speaker, I would think it sounds perfectly natural to hear someone say “I’m a Syrian expat”, but I would only use the term to describe “the Syrian expat community” (i.e. the Syrians that have come here and are relying on each other for support). If I were describing the same person, I would say “Syrian immigrant” because I’m not the one feeling the culture shock of being in a foreign land. (or I would use the term “refugee” which carries the implication that they’re here, but not by choice - they were forced out of their home)

“Immigrant” often also implies some sort of formal legal status, although in a looser sense it just means that you live in that country on a permanent basis. All immigrants are also expats, but not all expats are immigrants.

As others have pointed out here, while neither term is by itself positive or negative, “expat” will almost never be used in a negative sense, but “immigrant” can be used in a derogatory way, although it can also be neutral or positive depending on the speaker and context.