The lack of answers to this question may suggest that it's a tough one to answer. I think many of us 'experts' might say there is no difference but I think there's always something going on behind the scenes when we select one preposition over another, usually a subtle emphasis.
To me, the preposition 'at' suggests the anger travels in the direction of its object, perhaps suggesting shouting at or glaring at, for example.
Whereas the 'with' is perhaps a little less aggressive, maybe referring to a state of anger, not necessarily being shown to or aimed at the object of anger. According to this view, then:-
- angry with may be used to refer to a state of anger about something.
- angry at may emphasise a feeling of anger directed towards something.
Please be aware that these are merely my thoughts on the question and that I'm just offering my own way of understanding.
Interested in Language