How does this iPhone app measure the power flowing through an AC appliance cord?
Not having used said app I believe it is a master of assumptions and averages.

The far field of a transmission line with two opposite current conductors is essentially zero.  However the near field on anything other than a solid screen co-ax will always have a unbalanced component around it.

The maximum unbalance will be when the two current conductors are 'eclipsed' and the near field will be mostly due to the nearer conductor.

What I suggest for better trials is that you take a mains cable and carefully remove the outer covering  for 12 inches or so to separate the internal wires (do this with a spare kettle cord if you like).  Then run your tests with the phone placed next to just one of the live wires and see if it has repeatable results.

It is also important to remember that the magnetic field sensor in the phone should be a 3 axis device so it can determine the magnetic field 'direction' and infer the conductor placement to a degree and correct a bit.

Accuracy of 30% compared to a somewhat more predictable device I think is pretty phenomenal like you have found.  Getting better will obviously be much harder given the loose test conditions required. 

It does remind me of a totally passive DC ammeter that my friend had that was over 50 years old.  It was merely placed against a (single) conductor (such as the alternator charging lead in a car) and would deflect its needle to indicate the local induced magnetic field proportional to the current.  Orientation was important and it would not have read accurately if a return conductor was nearby, however the principal is pretty much the same.