"Homeless" has a kind of stigma to it (maybe you picture a dirty bum on the street), while "unhoused" doesn't because it's so new. It simply sounds like a category of analysis devoid of emotional judgment. (Like maybe they're unhoused because a hurricane just blew down their house, not because they don't have a job.)
If "unhoused" comes to replace "homeless", I'm sure we'll get another term to replace it, once "unhoused" has acquired the same stigma that "homeless" does now.
Though I've got to say that "unhoused" is a particularly uncreative term! Essentially a literal synonym of "homeless". It's like they didn't try at all this time...
My favorite offshoot of this is when I heard a Brit say, “don’t mind that guy, he’s currently experiencing Frenchness.”
Yup, plenty of people say that's the entire reason for the euphemism treadmill. You look good using the new terminology but nobody is helped.
And the people actually affected by the problem generally aren't consulted, and don't care at all about the terminology because it's the least of their problems.
So I would argue that being unhoused is a more accurate description of many people's experiences and feelings than being homeless.
You can't just solve the problem by sticking them into flats; that might technically make them not homeless anymore, but at the same time fixes basically none of their problems.