I think angry is more serious than indignant, and applies to many different situations.
e.g. screaming and shouting is angry, more than just indignant.
e.g. an angry letter could be crazy or rude (tomorrow maybe you regret writing it)
An indignant letter sounds more polite "Dear sir, I was very disappointed to hear that..."
As Andrea says, "indignant" also implies that there is a good reason: a reaction to something wrong or unjust (someone's opinion...).
e.g. I can be angry about missing my bus, but not indignant.
Then you tell me "you're so stupid to miss your bus" and I am indignant: "hey, everyone misses the bus sometimes, it doesn't make me stupid"