Some German verbes seem to have two different past participles.
e.g. hängen -> gehangen/gehängt
Can you choose whichever you want or is there any difference in meaning, application or regional use?
Actually, we have two slightly different verbs with different grammar here. One verb is hängen and is intransitive, the other is etwas (an einen Ort) hängen and transitive.
The intransitive verb is always conjugated as hängen – hing – gehangen. It means to hang somewhere in the way that e.g. a coat is hanging on a hanger or a picture is hanging on the wall.
Die Jacke hängt am Haken. → Die Jacke hing am Haken. → Die Jacke ist am Haken gehangen.
Das Bild hängt schief. → Das Bild hing schief. → Das Bild ist schief gehangen.[1]
Note that the place does not have to be specificied but that there is often an additional specifier present. If the place is specified with a Wechselpräposition, the noun following it is in dative case to signify a state.
The transitive verb is always conjugated as hängen – hängte – gehängt. It means to hang up something, i.e. it describes the process that leads to intransitive hängen. Therefore, it often turns up as aufhängen rather than pure hängen.
Ich hänge das Bild an die Wand. → Ich hängte das Bild an die Wand. → Ich habe das Bild an die Wand gehängt.
Ich hänge meine Jacke auf. → Ich hängte meine Jacke auf. → Ich habe meine Jacke aufgehängt.
Note that if the place is specified with a Wechselpräposition, the following noun is in accusative case to signify a movement towards there (of the object).
[1]: A number of verbs that use haben in perfect tense in northern Germany use sein in the South, Austria, South Tyrol and probably also Switzerland. Being from Bavaria, I use the southern-typical sein.