The main difference is choice. Simply speaking, a migrant is someone who chooses to move, and a refugee is someone who has been forced from their home. ... Migrants, on the other hand, may move for any number of reasons. Some of them move to be with family or for economic reasons
Here are the requirement to apply for refugee status to the U.S.: https://www.alllaw.com/articles/nolo/us-immigration/how-to-prepare-submit-refugee-application.html
Most of the people showing up at the Mexican/U.S. border want to move to the U.S. for economic reasons. They are not refugees.
Anyone who wants to apply to legally emigrate to the U.S., refugee or otherwise, must do so from their home country.