Disclaimer This post refers to the US only,
Let me first state that technically neither of these phrases really mean anything, theyre both ment to be as vague and non-descript as possible.
With the relatively recent influx of 24 hour, in-your-face news, many journalists and news outlets seem more interested in filling news feeds and getting clicks than actually dispersing important information. And with firearms being such a controversial subject, hard liners being on every side, misinformation runs rampant.
Assault rifle
Often mis-attributed to Hitler, the term "assault rifle" was actuallyvfirst used in the publication "FSTC-CW-07-03-70" in 1970, this publication was written by a civilian "department" of the DOD that was loosely affiliated with the US Army. It describes "short, compact, selective-fire weapons that fire a cartridge intermediate in power between submachinegun and rifle cartridges....Assault rifles have mild recoil characteristics and, because of this, are capable of delivering effective full-automatic fire at ranges up to 300 meters." (Source: http://031d26d.namesecurehost.com/gunfax/aw.htm)
To the average person this may seem legitimate and "official" however, this phrase does not appear in more "modern" publications nor it does not define what "rifle" and "submachine gun" cartridges are. These seemingly trivial pieces of information can actual make a surprisingly huge difference when you are referring to firearms when .001 in (.0254mm) Or a few grains (1 grain = 64.79mg) can mean the difference between a chamber bursting open or normal function.
Assault Weapon
While I havent been able to track down the origin for this term we do know it was used before 1994 when the federal "Assault weapons ban" was established. It currently has 7 varying definitions that generally refer to semi-automatic firearms with certain cosmetic, ergonomic, or even safety features in the case of a "barrel shroud" (essentially a heat shield for your hands) as well as normally naming certain makes/models. The states that have adopted their own version are California, Conneticut, Colorado, Hawaii, Massachusetts, New Jersey and New York.