The term "unhoused" arose because it sounds like a more clinical, technical word to describe the situation of someone who does not have reliable shelter/residence compared to "homeless," which has some emotional implications from the root word "home".
However, my soapbox opinion is that it's better to use the term homeless specifically BECAUSE it has emotional attachments, and all good people SHOULD feel emotional at the concept of homelessness. In my opinion, changing to the term "unhoused" is a way of sterilizing the horror of homelessness, and in effect, it increases people's apathy towards something that is extremely important.