What is a good way to distinguish aircraft noise from other sounds?
You need to train a classifier. The way I would approach it is to allow users to submit audio recordings through your app, and indicate which sections of the recordings contain the plane. This would enable you to quickly crowd source a lot of data with which to build a model. Then you would find the STFT and volume of the signal over time. Finally, train the classifier (such as an SVM or neural network) using the amplitude spectrum and volume features. Once you have trained the classifier, you can use it to detect planes by extracting the same features on live audio (STFT isn't computationally expensive, so it should be feasible to do on the phone).

Here's a backgrounder on Audio signal classification.