iOS cellular usage statistics inaccurate?

AT&T has an app that will tell you exactly what AT&T says you're using. The carrier is always going to charge you according the their numbers so, might as well use their app.

The app will also enable you to see what your son is using and doesn't require that you manually reset it every month.

There will sometimes be small differences. The phone reports data sent and received at the phone. However, sometimes there is a failure in communications (especially if the signal is weak) and data must be re-transmitted. The retransmission counts on the carrier's side, but is not recorded on the phone because iOS doesn't know about it.

However, a large difference is much harder to explain. One possibility is the timing; the phone records data when sent. However carriers (including AT&T) aggregate data and report it in a "chunk" periodically. AT&T specifically reports data for a "MTSO" (a collection of towers) when the phone moves out of range of the towers to a different location. So if he's at school using data, the total of that data usage will be reported when he leaves school and comes home. The network then starts accumulating home usage if he isn't on Wi-Fi, but doesn't report that until he leaves the home area. If he doesn't leave the home area the total of that usage will be reported between midnight and 3 AM (depending on time zone). In summary, there is no way to align the data usage times between the carrier's recording and the phone's recording. If you track for a longer period of time (like a week) they should be fairly close. But not for a day.

There is no way to get which apps use data from the AT&T site, because AT&T doesn't know that information, and you don't want them to, because they are rather "liberal" in how they share data they collect with third parties (see: AT&T, Sprint, Verizon to Stop Sharing Customer Location Data With Third Parties — Krebs on Security).

In Settings/Cellular it will list data by app. I reset it once a month at the start of the billing period, and sometimes take screen shots if I am tracking something. Over the long haul it's pretty accurate.