

It’s two less syllables to say “April Fourth” than “The Fourth of April”.
That’s about the only advantage it has.
Edit:
I was thinking about this grammatically. English is an Adjective first language where the modifying adjective goes before the base noun.
In my example, April is the adjective. It tells the reader what kind of Fourth it is.
It’s at least a kind of logic.
FYI, I added more in the base comment.