hi! there may be a few people who read Old English here, but this question might find more responses in a linguistics sub, e.g. /r/asklinguistics, /r/Linguistics (NB: in their stickied Q&A thread), or the very tempting-looking /r/OldEnglish

The thorn is for the unvoiced "th" while the eth is for the voiced "th". The way to tell the difference is to say the words "thin" and "then": the only difference between the two words is whether the "th" is voiced or not. Place your fingers on your voice box and say the two words, and you'll notice that with "then" the voice box vibrates with the "th" - that's an eth; with "thin" there is no vibration - that is the unvoiced thorn.

Edit: My source is my language mentor's user-friendly Old English Grammar: Robert E. Diamond (1920-1985), Old English Grammar and Reader (1970). The difference in use between the thorn and eth in Anglo-Saxon and in Old Norse was strictly determined by the sound of the "th"; they were not interchangeable.

Both letters may be used interchangeably. There is no difference in use between them.