I'm sure there are exceptions, but I've noted the following generic difference:
- "Раздаться" is a better choice if the sound is loud and harsh. E.g. "раздался гром", "раздались аплодисменты".
- "Донестись" is a better choice for quiet, barely audible sounds. Also if the source of the sound is distant and you want to make a point that the sound reached the listener. E.g. "донёсся шёпот". Collocations like "донёсся гром" and "донёсся выстрел" are also possible, but then they would imply "from a distance".
And of course, there are those nearly-idiomatic phrases where the other verb simply wouldn't work: "донеслись слухи".
Grammar-wise, they typically take different location adverbials:
- "раздаться (где?) в лесу", "раздаться (откуда?) из леса".
- "донестись (до кого?) до меня", "донестись (откуда?) из леса".
Update
It may help to bear in mind some lexical intuition behind those words:
the prefix "раз-" conveys a meaning of distribution (spread), whereas the prefix "до-" conveys a meaning of action/goal completion. That may explain why "раздаться" is better suited to answer "in what area?", and "донестись" is better suited to answer "to what recepient?", but not vice versa.