Assault Rifle = selective fire rifle or carbine firing intermediate or pistol cartridge (e.g. M16 or M4) instead of a battle rifle which fired a rifle cartridge (30.06 out of a Garand, 7.62 NATO out of a FAL, etc.)
Assault Weapon = semiauto rifle or carbine that happens to look kinda like a submachine gun or assault rifle firing an intermediate cartridge but the term sounds a lot scarier than just saying semiauto rifle or semiauto carbine (e.g. AR15 or M4gery) and therefore had a whole useless law ('94 AWB) built around the propaganda that somehow a semiauto rifle with a pistol grip, removable magazine and a flash hider was more dangerous than a blue and wood semiauto rifle firing the exact same round (AR15 vs Mini 14). Luckily, the law sun-setted, but the pejorative term lingers on as part of the common lexicon.
AW is a propaganda term while assault rifle is a military term.