Somewhere I got the naive idea that, in Old English, thorn represented the unvoiced "th" sound and eth represented the voiced "th" sound. A little digging has suggested to me that each of the characters could be used for each of the sounds. I'm wondering what rules governed their use in writing, and how they changed over time.
I agree @sumelic. I do think there is a subtle difference between the 'thorn' and 'eth'. Thorn is often used at the end of a word while eth is often in the middle of the word. This leads to a subtle different phonemic pronunciation.
I found this brief explanation with Old English examples: Also, ð and Ð (eth): Old English scribes could also represent the "th" sound with the letter ð (the capital letter version looks like a capital D with a short horizontal line: Ð). The letter is called "eth," pronounced so that it rhymes with the first syllable in the word "feather."