The distinction between hanged and hung is one of the odder ones in the language. I remember learning in high school that people are hanged, pictures are hung. There was never any explanation of why it was so; it simply was. It was years before I learned the strange and complicated history of these two words.
English has a few pairs of related verbs that are differentiated by their transitivity: lay/lie, rise/raise, and sit/set. Transitive verbs take objects; intransitive ones don’t. In each of these pairs, the intransitive verb is strong, and the transitive verb is weak. Strong verbs inflect for the preterite (simple past) and past participle forms by means of a vowel change, such as sing–sang–sung. Weak verbs add the -(e)d suffix (or sometimes just a -t or nothing at all if the word already ends in -t). So lie–lay–lain is a strong verb, and lay–laid–laid is weak. Note that the subject of one of the intransitive verbs becomes the object when you use its transitive counterpart. The book lay on the floor but I laid the book on the floor.
Historically hang belonged with these pairs, and it ended up in its current state through the accidents of sound change and history. It was originally two separate verbs (the Oxford English Dictionary actually says it was three—two Old English verbs and one Old Norse verb—but I don’t want to go down that rabbit hole) that came to be pronounced identically in their present-tense forms. They still retained their own preterite and past participle forms, though, so at one point in Early Modern English hang–hung–hung existed alongside hang–hanged–hanged.
Once the two verbs started to collapse together, the distinction started to become lost too. Just look at how much trouble we have keeping lay and lie separate, and they only overlap in the present lay and the past tense lay. With identical present tenses, hang/hang began to look like any other word with a choice between strong and weak past forms, like dived/dove or sneaked/snuck. The transitive/intransitive distinction between the two effectively disappeared, and hung won out as the preterite and past participle form.
The weak transitive hanged didn’t completely vanish, though; it stuck around in legal writing, which tends to use a lot of archaisms. Because it was only used in legal writing in the sense of hanging someone to death (with the poor soul as the object of the verb), it picked up the new sense that we’re now familiar with, whether or not the verb is transitive. Similarly, hung is used for everything but people, whether or not the verb is intransitive.
Interestingly, German has mostly hung on to the distinction. Though the German verbs both merged in the present tense into hängen, the past forms are still separate: hängen–hing–gehangen for intransitive forms and hängen–hängte–gehängt for transitive. Germans would say the equivalent of I hanged the picture on the wall and The picture hung on the wall—none of this nonsense about only using hanged when it’s a person hanging by the neck until dead.
The surprising thing about the distinction in English is that it’s observed (at least in edited writing) so faithfully. Usually people aren’t so good at honoring fussy semantic distinctions, but here I think the collocates do a lot of the work of selecting one word or the other. Searching for collocates of both hanged and hung in COCA, we find the following words:
The hanged words pretty clearly all hanging people, whether by suicide, as punishment for murder, or in effigy. (The collocations with burned were all about hanging and burning people or effigies.) The collocates for hung show no real pattern; it’s simply used for everything else. (The collocations with neck were not about hanging by the neck but about things being hung from or around the neck.)
So despite what I said about this being one of the odder distinctions in the language, it seems to work. (Though I’d like to know to what extent, if any, the distinction is an artifact of the copy editing process.) Hung is the general-use word; hanged is used when a few very specific and closely related contexts call for it.