In a nutshell, the 9mm is a pistol round and 5.56 is a rifle round and in almost all cases, a rifle round is more effective than pistol round.
More specifically:
- 9mm won't penetrate body armory but 5.56 will.
- Both rounds will penetrate standard home walls but the 5.56 is a wee bit safe because the walls will decelerate the 5.56 significantly more than the 9mm.
- The recoil of the 9mm AR is actually heavier (to me) because of the heavier bolt and heavier buffer.
- The 5.56 is going to be a lot louder indoors than a 9mm will.
- The 5.56 relies on velocity for lethality so the bbl length on a 5.56 AR is much more important than the bbl length of a 9mm AR.
- A carbine length 9mm AR will impart slightly more velocity on the round but not enough to make a noticeable difference in ballistics.
- A 9mm carbine is, for all intents and purposes, useless beyond 50 yds whereas the 5.56 is not.
- Accuracy with either caliber, within their useable ranges, is a moot point however, beyond 100 yds, the 9mm round will drop like a rock whereas the 5.56 is relatively flat out to 225 yds.
One last point to consider - entry teams across the country have switched or are switching from pistol caliber carbines/SMGs to M4s/short bbl ARs in droves. As to why they are doing that, please review my previous points.
9mm ARs are fun. They're cheap to shoot (for the most part), are better to suppress (silencers), can be SBR'd without affecting their effectiveness and can be shot at any indoor range (some ranges do not allow rifle calibers indoors) but they are not 5.56 ARs.
Regarding your comment on "the .223 round is a bit small for some applications", if that is the case, then you should consider larger rifle calibers such as 6.8 SPC, 7.62x39 or .308. The .223/5.56 may be small but when implemented properly, it is considerably more lethal than 9mm, shot placement not withstanding of course.
Finally, ask yourself "Would I rather get shot with a pistol round or rifle round?". I would venture to guess that you would chose the former over the latter if you had to choose.